Use of matrix partitioning to determine the wigner rotation matrix (thomas rotation matrix) in rodrigues form specifying the wigner angle (thomas angle) and the axis by multiplication of three lorentz boost matrices

ABSTRACT

For two given restricted Lorentz transformation matrices ALη and BLη with identical first column vectors the calculation of the rotation matrix Rη=(ALη)−1 BLη can according to the invention be simplified by matrix partitioning. Using this method the calculation of the Wigner rotation matrix (Thomas rotation matrix) WRη, which is for two given Lorentz boost matrices B1η, B2η defined by (B1η)(B2η)=(B3η)(WRη) with B3η being a further Lorentz boost matrix, can be simplified by making the identifications ALη=B3η and BLη=(B1η)(B2η). Since the matrix formed by the four four-vectors of a local frame of a timelike worldline in 4-dimensional Minkowski space can be interpreted as a proper time dependent restricted Lorentz transformation matrix, said method can also be used to simplify the calculation of the rotation matrix linking two different local frames of a given timelike worldline.

FIELD OF THE INVENTION

It is recommended to read this U.S. patent application Ser. No. ______in the form of the pdf-document originally filed, which can bedownloaded underhttps://patentcenter.uspto.gov/applications/XXXXXXXX/ifw/docs (pdf-linkin the line with the document description ‘specification’), since theapplication as published by the USPTO is presumably riddled withmistakes.

This invention relates to the manipulation of matrices representingrestricted Lorentz transformations [4, chapter 6 on p. 167, inparticular subchapter “6.3.3. Restricted Lorentz Group” on p. 174] infour-dimensional Minkowski space.

In this application we use for the Minkowski metric the convention

$\eta = \begin{pmatrix}{- 1} & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{pmatrix}$

Greek indices run from 0 to 3, latin indices run from 1 to 3. Capitalletters L, R and B denote contravariant tensors of rank 2 with matrixcomponents L^(μν), R^(μν), B^(μν). The products Lη, Rη and Bη denotemixed tensors of rank 2 with matrix components L^(μ) _(ν), R^(μ) _(ν),B^(μ) _(ν). The products ηLη, ηRη and ηBη denote covariant tensors ofrank 2 with matrix components L_(μν), R_(μν), B_(μν).

A point · between four-vectors denotes always the Minkowski pseudoscalar product (a·u=a^(T)ηu=a_(ν)u^(ν)), a point · betweenthree-dimensional vectors denotes always a normal Euclidian scalarproduct (a·u=a^(T)u) and a point · between a matrix and another matrixor a vector denotes always matrix multiplication, however the latterpoint is most often omitted. Throughout the application documentsthree-dimensional quantities are always denoted by bold symbols like r,u, a, ω, R, L, B in contrast to the associated four-dimensionalquantities r, u, a, R, L, B. Three-dimensional vectors with a hat{circumflex over ( )} such as {circumflex over (ω)} denote always unitvectors.

BACKGROUND OF THE INVENTION

Textbooks on special relativity like [4, equ.(4.42) on p. 108,equ.(4.41) on p. 107, equ.(4.19) on p. 102, equ.(2.12) on p. 35 andremark 2.9 on p. 36] or [8, equ.(6.6) on p. 164 and equ.(4.23) on p.118] or [7, equ.(11.36) on p. 532 and equ.(11.17) on p. 525] or [5,equ.(17.36) on p. 373] disclose that if a massive particle is movingwith constant speed v=(υ¹, υ², υ³)^(T) relative to an inertial framewith time coordinate t, then the particle is represented in4-dimensional Minkowski space by the timelike worldline

$\begin{matrix}{{r(\tau)} = {{\begin{pmatrix}1 \\\frac{v}{c}\end{pmatrix} \cdot {t(\tau)}} = {{\begin{pmatrix}1 \\\frac{v}{c}\end{pmatrix} \cdot \frac{1}{\underset{\gamma}{\underset{︸}{\sqrt{1 - \left( \frac{v}{c} \right)^{2}}}}} \cdot \tau} = {\begin{pmatrix}1 \\\frac{v}{c}\end{pmatrix} \cdot \frac{1}{\underset{\gamma}{\underset{︸}{\sqrt{1 - {\sum\limits_{i = 1}^{3}\left( \frac{v^{i}}{c} \right)^{2}}}}}} \cdot \tau}}}} & (1)\end{matrix}$

with τ being the proper time of the particle and with c denoting thespeed of light. In line with [4, equ. 2.12 on p. 35] we define thefour-velocity u of the particle by

$\begin{matrix}{u = {{\frac{1}{c}\frac{dr}{d\tau}} = {\begin{pmatrix}u^{0} \\u\end{pmatrix} = {\begin{pmatrix}\sqrt{1 + u^{2}} \\u\end{pmatrix} = {\begin{pmatrix}u^{0} \\u^{1} \\u^{2} \\u^{3}\end{pmatrix} = {{\gamma\begin{pmatrix}1 \\\frac{v}{c}\end{pmatrix}} = {\gamma\begin{pmatrix}1 \\\frac{v^{1}}{c} \\\frac{v^{2}}{c} \\\frac{v^{3}}{c}\end{pmatrix}}}}}}}} & (2)\end{matrix}$

such that

u·u=u ^(T) ηu=u _(μ) u ^(μ)=η_(μν) u ^(ν) u ^(μ)=−1  (3)

Other authors like [8, equ.(6.6) on p. 164], [7, equ.(11.36) on p. 532],[5, line between equ.(17.30) and equ.(17.31) on p. 372 or equ.(17.36) onp. 373] or [1, p. 819, left col., paragraph preceding equ.(5)] definethe four-velocity by

${u = \frac{dr}{d\tau}},$

in which case equ.(3) assumes the form u·u=−c²[8, Exercise 6.2.1. on p.164], [5, equ.(17.20) on p. 369] and [1, p. 819, left col., paragraphfollowing equ.(5)]. Also in the latter case equ.(3) remains valid if onechooses natural units with c=1.

A Lorentz transformation matrix Lη=(L^(μ) _(ν)) is a 4×4-matrixrepresenting a mixed tensor, which fulfills one of the four followingequivalent conditions ([4, p. 171]):

(Lηb)·(Lηc)=(Lηb)^(T)η(Lηc)=b ^(T) ηc=b·c for all four-vectors b,c  (4)

⇔(Lη)^(T)η(Lη)=η  (5)

⇔(Lη)⁻¹=η(Lη)^(T)η  (6)

⇔η_(αβ) L ^(α) _(μ) L _(ν) ^(β)=η_(μν)  (7)

A Lorentz transformation matrix, which transforms coordinates in a righthanded Minkowski-orthogonal basis moving along a straight timelikeworldline as defined in equ.(1) into coordinates in a right-handedMinkowski-orthogonal basis at rest, has to be a restricted Lorentztransformation matrix, i.e. it has to obey additionally the twofollowing conditions ([4, equ.(6.18) on p. 174]):

det(Lη)=1 and L ⁰ ₀≥1  (8)

These conditions exclude time and space inversions and ensure that theparticle is moving in the future direction [4, p. 16].

Examples of restricted Lorentz transformations are purely spatialrotations

$\begin{matrix}{{{R\eta} = {{{\begin{pmatrix}1 & 0^{T} \\0 & R\end{pmatrix}{with}R} \in {{{SO}\left( {3,} \right)}{and}0}}:=\begin{pmatrix}0 \\0 \\0\end{pmatrix}}},} & (9)\end{matrix}$ (Rη)^(T) = (Rη)⁻¹ ⇔ R^(T) = R⁻¹anddet (Rη) = 1 ⇔ det R = 1

and Lorentz boosts, which are also called pure Lorentz transformationsor special Lorentz transformations (some authors denote only boosts asLorentz transformations and use different expressions for othertransformations of the Lorentz group or of the restricted Lorentzgroup). The matrix of a Lorentz boost can be brought in the followingform ([3, equ.(3-46) on p. 66 and equ.(3-33) on p. 53], [4, equ.(6.72)on p. 198]):

$\begin{matrix}{{B\eta} = {\begin{pmatrix}u^{0} & u^{T} \\u & B\end{pmatrix} = {{\begin{pmatrix}u^{0} & u^{T} \\u & {1 + \frac{{uu}^{T}}{u^{0} + 1}}\end{pmatrix}{with}1}:=\begin{pmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{pmatrix}}}} & (10)\end{matrix}$ $\begin{matrix}{\left( {B\eta} \right)^{- 1} = {\begin{pmatrix}u^{0} & {- u^{T}} \\{- u} & B^{T}\end{pmatrix} = \begin{pmatrix}u^{0} & {- u^{T}} \\{- u} & {1 + \frac{{uu}^{T}}{u^{0} + 1}}\end{pmatrix}}} & \text{(11)}\end{matrix}$ $\begin{matrix}{B^{- 1}\overset{\overset{{annex}1}{\downarrow}}{=}{1 - \frac{{uu}^{T}}{u^{0}\left( {u^{0} + 1} \right)}}} & \text{(12)}\end{matrix}$

Note that other authors define the boost and the inverse boost exactlythe other way around ([8, equ.(1.45) on p. 25], [7, equ.(11.98) on p.547]).

It is further known from textbooks ([4, chapter “6.5. PolarDecomposition” on p. 191-193]), that any restricted Lorentztransformation matrix Lη can be written in a unique way as the productof a boost matrix Bη and a rotation matrix Rη:

Lη=(Bη)(Rη)  (13)

The restricted Lorentz transformation matrices form with the matrixmultiplication a group in the mathematical sense ([4, p. 174, middleparagraph]), i.e. the product of two restricted Lorentz transformationmatrices yields always another restricted Lorentz transformation matrix.But the Lorentz boost matrices do not form a subgroup in themathematical sense, i.e. the product of two Lorentz boost matrices B₁η,B₂η does in general not yield another Lorentz boost matrix, but arestricted Lorentz transformation matrix, which can as said above bedecomposed in the product of another Lorentz boost matrix B₃η and arotation matrix ^(W)Rη called Wigner rotation matrix or Thomas rotationmatrix ([4, equ.(6.100) on p. 209]):

$\begin{matrix}{\left. \left. {{\left( {B_{1}\eta} \right)\left( {B_{2}\eta} \right)} = {\left( {B_{3}\eta} \right)\left( {}^{W}R\eta \right.}} \right)\Leftrightarrow{\begin{pmatrix}u_{1}^{0} & u_{1}^{T} \\u_{1} & B_{1}\end{pmatrix}\begin{pmatrix}u_{2}^{0} & u_{2}^{T} \\u_{2} & B_{2}\end{pmatrix}} \right. = {\left. {\begin{pmatrix}u_{3}^{0} & u_{3}^{T} \\u_{3} & B_{3}\end{pmatrix}\begin{pmatrix}1 & 0^{T} \\0 & {\,^{W}R}\end{pmatrix}}\Leftrightarrow\begin{pmatrix}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}} & {{u_{1}^{0}u_{2}^{T}} + {u_{1}^{T}B_{2}}} \\{{u_{2}^{0}u_{1}} + {B_{1}u_{2}}} & {{u_{1}u_{2}^{T}} + {B_{1}B_{2}}}\end{pmatrix} \right. = \begin{pmatrix}u_{3}^{0} & u_{3}^{T} & {\,^{W}R} \\u_{3} & B_{3} & {\,^{W}R}\end{pmatrix}}} & (14)\end{matrix}$

The third Lorentz boost matrix B₃η can easily be determined. Firstlyfrom the preceding equation one can directly read that the first columnvector of the third Lorentz boost matrix B₃η is given by

$\begin{matrix}{\begin{pmatrix}u_{3}^{0} \\u_{3}\end{pmatrix} = \begin{pmatrix}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}} \\{{u_{2}^{0}u_{1}} + {B_{1}u_{2}}}\end{pmatrix}} & (15)\end{matrix}$

Secondly one can see from equ.(10) that a Lorentz boost matrix is fullydetermined by its first column vector, i.e.

${B_{3}\eta} = {\begin{pmatrix}u_{3}^{0} & u_{3}^{T} \\u_{3} & B_{3}\end{pmatrix} = \begin{pmatrix}u_{3}^{0} & u_{3}^{T} \\u_{3} & {1 + \frac{u_{3}u_{3}^{T}}{u_{3}^{0} + 1}}\end{pmatrix}}$

With the third Lorentz boost matrix B₃η thus given the Wigner rotationmatrix ^(W)Rη can be calculated by resolving equ.(14) for the Wignerrotation matrix:

$\begin{matrix}{{{\,^{W}R}\eta} = {\begin{pmatrix}1 & 0^{T} \\0 & {\,^{W}R}\end{pmatrix} = {{{\underset{{({{\,^{A}L}\eta})}^{- 1}}{\underset{︸}{\left( {B_{3}\eta} \right)^{- 1}}}\underset{{\,^{B}L}\eta}{\underset{︸}{\left( {B_{1}\eta} \right)\left( {B_{2}\eta} \right)}}} == {\begin{pmatrix}u_{3}^{0} & u_{3}^{T} \\u_{3} & B_{3}\end{pmatrix}^{- 1}\begin{pmatrix}u_{1}^{0} & u_{1}^{T} \\u_{1} & B_{1}\end{pmatrix}\begin{pmatrix}u_{2}^{0} & u_{2}^{T} \\u_{2} & B_{2}\end{pmatrix}} == {\begin{pmatrix}u_{3}^{0} & u_{3}^{T} \\u_{3} & B_{3}\end{pmatrix}^{- 1}\begin{pmatrix}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}} & {{u_{1}^{0}u_{2}^{T}} + {u_{1}^{T}B_{2}}} \\{{u_{2}^{0}u_{1}} + {B_{1}u_{2}}} & {{u_{1}u_{2}^{T}} + {B_{1}B_{2}}}\end{pmatrix}}} =}}} & (16)\end{matrix}$ $\begin{matrix}{= {\underset{{({{\,^{A}L}\eta})}^{- 1}}{\underset{︸}{\begin{pmatrix}u_{3}^{0} & u_{3}^{T} \\u_{3} & B_{3}\end{pmatrix}^{- 1}}}\underset{{\,^{B}L}\eta}{\underset{︸}{\begin{pmatrix}u_{3}^{0} & {{u_{1}^{0}u_{2}^{T}} + {u_{1}^{T}B_{2}}} \\u_{3} & {{u_{1}u_{2}^{T}} + {B_{1}B_{2}}}\end{pmatrix}}}}} & (17)\end{matrix}$

However this seemingly simple task turns out to be complex when donemanually and was in fact considered to be prohibitively complex fordecades ([9, abstract], [2, p. 58, 2nd para.]), such that other methodsto determine the Wigner rotation angle ^(W)

and the Wigner rotation axis ^(W){circumflex over (ω)} have beendeveloped (see for

example [4, chapter “6.7.2 Thomas Rotation” on p. 206-211 and chapter“6.7.3 Thomas Rotation Angle” on p. 212-215, in particular the“Historical note” on p. 215], [10, in particular chapter “1.Introduction”], [8, p. 180], [9] and [11]).

With the Wigner angle ^(W)

and the rotation axis ^(W){circumflex over (ω)} given the Wignerrotation matrix ^(W)R can also be brought in the following standard formusing the Rodrigues formula [6, p. 393], which defines an arbitraryrotation matrix R in terms of the rotation angle

and the unit vector {circumflex over (ω)} pointing in the direction ofthe rotation axis:

$\begin{matrix}{R = {{{\left( {\sin\vartheta} \right)\left\lbrack \hat{\omega} \right\rbrack}_{\times} + 1 + {\left( {1 - {\cos\vartheta}} \right)\left( \left\lbrack \hat{\omega} \right\rbrack_{\times} \right)^{2}}} =}} & (18)\end{matrix}$ $\begin{matrix}{= {\underset{\frac{1}{2}{({R - R^{T}})}}{\underset{︸}{{\left( {\sin\vartheta} \right)\left\lbrack \hat{\omega} \right\rbrack}_{\times}}} + \underset{\frac{1}{2}{({R - R^{T}})}}{\underset{︸}{{\left( {1 - {\cos\vartheta}} \right)\hat{\omega}{\hat{\omega}}^{T}} + {\cos\vartheta 1}}}}} & \text{(19)}\end{matrix}$

with the unit matrix 1 being defined in equ.(10) and with the skewsymmetric matrix [{circumflex over (ω)}]_(x) being defined by

$\left\lbrack \hat{\omega} \right\rbrack_{\times}:={{\begin{pmatrix}0 & {- {\hat{\omega}}_{3}} & {\hat{\omega}}_{2} \\{\hat{\omega}}_{3} & 0 & {- {\hat{\omega}}_{1}} \\{- {\hat{\omega}}_{2}} & {\hat{\omega}}_{1} & 0\end{pmatrix}{\overset{\overset{{{{\lbrack{6,{{fact}4.12{\text{.1}.i}}}})}{on}p\text{.384}}\rbrack}{\downarrow}}{\left. \Rightarrow \right.}\left\lbrack \hat{\omega} \right\rbrack}_{\times}^{2}} = {{\hat{\omega}{\hat{\omega}}^{T}} - 1}}$

such that for any three dimensional vector a the following applies [6,fact 4.12.1. viii) on p. 384]:

[{circumflex over (ω)}]_(x) ·a={circumflex over (ω)}×a

Equ.(19) shows that an easy way to determine the rotation angle

and the rotation axis {circumflex over (ω)} of an arbitrary rotationmatrix R is to determine the antisymmetric part

$\frac{1}{2}\left( {R - R^{T}} \right)$

except if the antiymmetric part turns out to be equal to [0]_(x). In thelatter case the rotation angle is either

=0 and the rotation axis undetermined (and the rotation matrix the unitymatrix R=1) or the rotation angle is

=π and

=π=R=2{circumflex over (ω)}{circumflex over (ω)}^(T)−1

such that the three components of the rotation axis {circumflex over(ω)} can in this case be determined as the three square roots of thethree diagonal elements of the matrix

$\frac{1}{2}{\left( {R + 1} \right).}$

Equ.(19) shows further that another way to determine the rotation angle

of an arbitrary rotation matrix R is to evaluate the trace of therotation matrix, which requires only the symmetric part:

$\begin{matrix}{{{tr}R} = {{{tr}\left\lbrack {\frac{1}{2}\left( {R + R^{T}} \right)} \right\rbrack} = {{{\left( {1 - {\cos\vartheta}} \right)\left( {{\hat{\omega}}_{1}^{2} + {\hat{\omega}}_{2}^{2} + {\hat{\omega}}_{3}^{2}} \right)} + {3\cos\vartheta}} = {1 + {2\cos\vartheta}}}}} & (20)\end{matrix}$

At least one of the first to determine the Wigner angle and the rotationaxis by simply evaluating equ.(16) appears to be D. E. Fahnline [1,abstract and chapter “IV. DECOMPOSITION OF THE PRODUCT OF TWO PURELORENTZ TRANSFORMATIONS”]. He tackled the problem using a covariant formof the Lorentz boost, which he introduced in said article and which hasmeanwhile found its way into at least one textbook ([8, chapter “15 TheCovariant Lorentz Transformation”]; however the very same textbookderives the Wigner angle not by the method disclosed in [1], see [8, p.180]). Fahnline obtains the 3-dimensional Wigner rotation matrix ^(W)Rin the form of a linear combination of a unity matrix and three dyads([1, equ.(30) on p. 820]) and illustrates the calculation path. Fahnlinethen uses equ.(20) above to determine the Wigner angle ^(W)

([1, p. 821, first paragraph]). The eigenvector of the Wigner rotationmatrix ^(W)R and thus the Wigner rotation axis ^(W){circumflex over (ω)}can be directly read from the matrix due to its simple structure.Fahnline does not transform the Wigner rotation matrix ^(W)R in theRodrigues form of equ.(18).

According to [2] Ungar derived the Wigner rotation matrix ^(W)R in theRodrigues form of equ.(18) likewise by evaluating equ.(16), but providedonly the final result ([2, equ.(14) on p. 63 and equ.(15a), (15b), (16)on p. 64 as well as equ.(4) on p. 61]) without giving intermediateresults. Fahnline [1] is cited by Ungar [2, reference 2 on p. 87].

SUMMARY OF THE INVENTION

The problem to be solved by this invention is to provide an alternativeway to evaluate the right hand side of equ.(16). Instead of using thecovariant form of the Lorentz boost as done by Fahnline we use apartitioned form of the Lorentz boost matrix. Moreover we use thefour-velocity u=(u⁰ u)^(T) instead of the Lorentz factor γ and the speedv/c, which simplifies the notation.

The problem of determining the Wigner rotation matrix ^(W)R according toequ.(16) can be generalised in the following way: the first columnvectors of the two Lorenz transformation matrices ^(A)Lη=B₃η and^(B)Lη=(B₁η) (B₂η) defined in equ.(16) are identical as can be seen inequ.(17). The more general problem is thus to determine the rotationmatrix Rη linking two restricted Lorentz transformation matrices ^(A)Lηand ^(B)Lη with identical first column vectors:

Rη=(^(A) Lη)^(−1B) Lη

This general problem is solved as defined in claim 1 by using matrixpartitioning. Advantageous embodiments are defined in dependent claims2-4.

BRIEF DESCRIPTION OF THE DRAWINGS

Not Applicable

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

In Annex 2.1. (or equivalently in annex 2.2.) we show that anyrestricted Lorentz transformation matrix Lη can be partitioned in thefollowing way:

$\begin{matrix}{{L\eta} = \begin{pmatrix}u^{0} & \left( {\frac{1}{u^{0}}L^{T}u} \right)^{T} \\u & L\end{pmatrix}} & (21)\end{matrix}$ with $\begin{matrix}{{LL}^{T} = {\left. {1 + {uu}^{T}}\Leftrightarrow L^{- 1} \right. = {L^{T}\left( {1 - \frac{{uu}^{T}}{\left( u^{0} \right)^{2}}} \right)}}} & (22)\end{matrix}$ and $\begin{matrix}{{\det L} = u^{0}} & \text{(23)}\end{matrix}$

Note that from equ.(22) and det(1+uu^(T))=1+u² ([6, fact 3.21.1. on p.351]) one can easily derive |det L|=u⁰, but only equ.(23) ensures thatdet(Lη)=1 (equ.(8) above) is fulfilled as shown in said annex 2.1. (andequivalently in annex 2.2). In annex 2.1. the three preceding equationsare proven using equ.(6) above, while in annex 2.2. the three precedingequations are proven using equ.(13), i.e. annexes 2.1. and 2.2. areequivalent.

With this partitioned form it is easy to calculate the rotation matrixlinking two restricted Lorentz transformation matrices with identicalfirst column vector, i.e. with identical four-velocity:

$\begin{matrix}{{{\,}^{A}L\eta} = {{\begin{pmatrix}u^{0} & \left( {\frac{1}{u^{0}}{\,^{A}L}{\,^{T}u}} \right)^{T} \\u & {\,^{A}L}\end{pmatrix}{and}{\,^{B}L}\eta} = \begin{pmatrix}u^{0} & \left( {\frac{1}{u^{0}}{\,^{B}L}{\,^{T}u}} \right)^{T} \\u & {\,^{B}L}\end{pmatrix}}} & (24)\end{matrix}$ ⇓${{\,^{B}L}\eta} = {\left. {\left( {{\,^{A}L}\eta} \right)\left( {R\eta} \right)}\Leftrightarrow\begin{pmatrix}u^{0} & \left( {\frac{1}{u^{0}}{\,^{B}L}{\,^{T}u}} \right)^{T} \\u & {\,^{B}L}\end{pmatrix} \right. = {{\begin{pmatrix}u^{0} & \left( {\frac{1}{u^{0}}{\,^{A}L}{\,^{T}u}} \right)^{T} \\u & {\,^{A}L}\end{pmatrix}\begin{pmatrix}1 & 0 \\0 & R\end{pmatrix}} = \text{ }{\left. \begin{pmatrix}u^{0} & \left( {\frac{1}{u^{0}}R^{T}{\,^{A}L}{\,^{T}u}} \right)^{T} \\u & {\,^{A}{LR}}\end{pmatrix}\Rightarrow{\,^{B}L} \right. = {\left. {\,^{A}{LR}}\Leftrightarrow R \right. = {{{\,^{A}L^{- 1}}{\,^{B}L}}\overset{\overset{{equ}.{(40)}}{\downarrow}}{=}\text{ }{{\,^{A}L^{T}}\left( {1 - {\frac{1}{\left( u^{0} \right)^{2}}{uu}^{T}}} \right){\,^{B}L}}}}}}}$$\begin{matrix}{\left. \Rightarrow{R\eta} \right. = {\begin{pmatrix}1 & 0 \\0 & R\end{pmatrix} = {{{\left( {L_{A}\eta} \right)^{- 1}L_{B}\eta} == \begin{pmatrix}1 & 0^{T} \\0 & {{\,^{A}L^{T}}\left( {1 - {\frac{1}{\left( u^{0} \right)^{2}}{uu}^{T}}} \right){\,^{B}L}}\end{pmatrix}}\overset{\overset{{equ}.{(40)}}{\downarrow}}{=}\begin{pmatrix}1 & 0^{T} \\0 & {{\,^{A}L^{- 1}}{\,^{B}L}}\end{pmatrix}}}} & (25)\end{matrix}$

This formula and the simple structure of B (equ.(10)) simplify thecalculation of the Wigner rotation matrix (Thomas rotation matrix) asshown in the following. From equ.(17), (24) and (25) follows

$\begin{matrix}{{\,^{W}R} = {{B_{3}^{- 1}\left( {{u_{1}u_{2}^{T}} + {B_{1}B_{2}}} \right)} = {\overset{\overset{{equ}.{(12)}}{\downarrow}}{=}\text{ }{{\left\lbrack {1 - \frac{u_{3}u_{3}^{T}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} \right\rbrack\left( {{u_{1}u_{2}^{T}} + {B_{1}B_{2}}} \right)} = {\overset{\overset{{equ}.{(15)}}{\downarrow}}{=}\text{ }{{\left\lbrack {1 - \frac{\left( {{u_{2}^{0}u_{1}} + {B_{1}u_{2}}} \right) \cdot \left( {{u_{2}^{0}u_{1}} + {B_{1}u_{2}}} \right)^{T}}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}} \right\rbrack\left( {{u_{1}u_{2}^{T}} + {B_{1}B_{2}}} \right)} = {\overset{\overset{{equ}.{(10)}}{\downarrow}}{=}\text{ }{{\left\lbrack {1 - \frac{\left( {{u_{2}^{0}u_{1}} + {\left( {1 + \frac{u_{1}u_{1}^{T}}{u_{1}^{0} + 1}} \right)u_{2}}} \right) \cdot \left( {{u_{2}^{0}u_{1}} + {\left( {1 + \frac{u_{1}u_{1}^{T}}{u_{1}^{0} + 1}} \right)u_{2}}} \right)^{T}}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}} \right\rbrack \cdot \cdot \text{ }\left\lbrack {{u_{1}u_{2}^{T}} + {\left( {1 + \frac{u_{1}u_{1}^{T}}{u_{1}^{0} + 1}} \right)\left( {1 + \frac{u_{2}u_{2}^{T}}{u_{2}^{0} + 1}} \right)}} \right\rbrack} =}}}}}}}} & (26)\end{matrix}$ $\begin{matrix}{{\overset{\overset{{annex}3}{\downarrow}}{=}{1 + {u_{1}u_{1}^{T}{\frac{1 - u_{2}^{0}}{\left( {u_{1}^{0} + 1} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}++}}}}\text{ }{u_{2}u_{2}^{T}{\frac{1 - u_{1}^{0}}{\left( {u_{2}^{0} + 1} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}++}}\text{ }{{{u_{1}u_{2}^{T}\frac{{2u_{1}^{T}u_{2}} + {\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}} + {{- u_{2}}u_{1}^{T}\frac{1}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1}}} =}} & (27)\end{matrix}$ $\begin{matrix}{\overset{\overset{{annex}5}{\downarrow}}{=}{{1 - {{{\frac{u_{1}^{0} + u_{2}^{0} + \overset{u_{3}^{0}}{\overset{︷}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}}} + 1}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)\left( {\underset{u_{3}^{0}}{\underset{︸}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}}} + 1} \right)}\left\lbrack {u_{1} \times u_{2}} \right\rbrack}_{\times}++}{\frac{1}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)\left( {\underset{u_{3}^{0}}{\underset{︸}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}}} + 1} \right)}\left\lbrack {u_{1} \times u_{2}} \right\rbrack}_{\times}^{2}}} =}} & (28)\end{matrix}$ $\begin{matrix}{\overset{\overset{{equ}.{(2)}}{\downarrow}}{=}{1 - {{{\frac{\gamma_{1} + \gamma_{2} + \overset{\gamma_{3}}{\overset{︷}{{\gamma_{1}\gamma_{2}} + {\gamma_{1}\gamma_{2}\frac{v_{1}^{T}v_{2}}{c^{2}}}}} + 1}{\left( {\gamma_{1} + 1} \right)\left( {\gamma_{2} + 1} \right)\left( {\underset{\gamma_{3}}{\underset{︸}{{\gamma_{1}\gamma_{2}} + {\gamma_{1}\gamma_{2}\frac{v_{1}^{T}v_{2}}{c^{2}}}}} + 1} \right)}\left\lbrack {\frac{\gamma_{1}v_{1}}{c} \times \frac{\gamma_{2}v_{2}}{c}} \right\rbrack}_{\times}++}{\frac{1}{\left( {\gamma_{1} + 1} \right)\left( {\gamma_{2} + 1} \right)\left( {\underset{\gamma_{3}}{\underset{︸}{{\gamma_{1}\gamma_{2}} + {\gamma_{1}\gamma_{2}\frac{v_{1}^{T}v_{2}}{c^{2}}}}} + 1} \right)}\left\lbrack {\frac{\gamma_{1}v_{1}}{c} \times \frac{\gamma_{2}v_{2}}{c}} \right\rbrack}_{\times}^{2}}}} & (29)\end{matrix}$

For the last step we used the relation

$u_{i} = {\begin{pmatrix}u_{i}^{0} \\u_{i}\end{pmatrix} = {\gamma_{i}\begin{pmatrix}1 \\\frac{v_{i}}{c}\end{pmatrix}}}$

for i=1, 2, 3 from equ.(2) above.

It is already apparent from equ.(26) that the expression for the Wignerrotation matrix is the sum of a unity matrix and a linear combination ofdyads and thus has the general structure of equ.(27), but thecalculation of the scalar factors associated with each dyad is lengthyas detailed in annex 3. However this calculation is straight forward andinvolves only school mathematics. Equ.(27) is the form in which theWigner rotation matrix was derived by Fahnline [1, equ.(30) on p. 820]as detailed in annex 6. From this form it is already apparent that u₁×u₂is a proper vector of the Wigner rotation matrix with proper value 1 andthus points into the direction of the axis of rotation ([1, p. 821, leftcol., penultimate paragraph]). Fahnline determined the Wigner rotationangle using equ.(20), i.e. using only the symmetric part of the Wignerrotation matrix ([1, p. 821, left col., first paragraph]). Thetransformation from equ.(27) to equ.(28) resides essentially in theseparation of the Wigner rotation matrix into the symmetric and theantisymmetric part and in the use of the relation [u₁×u₂]_(x)=u₂u₁^(T)−u₁u₂ ^(T)[6, fact 4.12.1 xi) on p. 384] as detailed in annex 5.Finally in the form of equ.(29) the Wigner rotation matrix was derivedby Ungar [2, equ.(4), (14), (15a), (15b), (16)].

Comparing equ.(28) with the standard Rodrigues form of a rotation inequ.(18) above yields the following expressions for the sine and thecosine of the Wigner rotation angle ^(W)

:

$\begin{matrix}{{\sin{\,^{W}\vartheta}} = {- \frac{\left( {u_{1}^{0} + u_{2}^{0} + \overset{u_{3}^{0}}{\overset{︷}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}}} + 1} \right)\sqrt{\left( {u_{1} \times u_{2}} \right)^{2}}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)\left( {\underset{u_{3}^{0}}{\underset{︸}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}}} + 1} \right)}}} & (30)\end{matrix}$ $\begin{matrix}{{\cos{\,^{W}\vartheta}} = {{1 - \frac{\left( {u_{1} \times u_{2}} \right)^{2}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)\left( {\underset{u_{3}^{0}}{\underset{︸}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}}} + 1} \right)}} =}} & (31)\end{matrix}$ $\begin{matrix}{\overset{\overset{{annex}7}{\downarrow}}{=}{\frac{\left( {u_{1}^{0} + u_{2}^{0} + \overset{u_{3}^{0}}{\overset{︷}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}}} + 1} \right)^{2}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)\left( {\underset{u_{3}^{0}}{\underset{︸}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}}} + 1} \right)} - 1}} & (32)\end{matrix}$

Equ.(30) is identical to the ‘Stapp formula’ ([4, equ.(6.118) on p.214]) and equ.(32) is identical to equ.(6.115) on p. 213 of textbook[4]. In annex 8 it is shown that the identity cos^(2 W)

+sin^(2 W)

=1 holds.

The method to calculate the rotation matrix linking two restrictedLorentz transformation matrices with identical first column vector usingequ.(24) as claimed in claim 1 can also be used to simplify thecalculation of the rotation matrix linking to different local frames ofthe same timelike worldline, since the mixed tensor matrix, which isformed by using the four four-vectors of a local frame of a timelikeworldline as column vectors, is a proper time T dependent Lorentztransformation matrix with the first column four-vector being formed bythe four-velocity u(τ)=(u⁰(τ)u(τ))^(T) of the timelike worldline [4,chapter “3.4.1 Local Frame of an Observer” on pages 76-78] or [12, para.0005-0010 on pages 3,4]. In particular two local frames for the sametimelike worldline have the same first column four-vector, such that themethod of claim 1 can be applied.

For example in [12, equ.(44) on p. 18 and equ.(53) on p. 22] two localframes L̆η, L̊η for the same timelike worldline with four-velocity u=(u⁰u)^(T) and four acceleration

$a = {\frac{du}{d\tau} = \left( {a^{0}a} \right)^{T}}$

(the explicit dependance on proper time τ is omitted for clarity) aredefined as follows:

${\overset{\smile}{L}\eta}:=\begin{pmatrix}u^{0} & \frac{a^{0}}{\sqrt{a^{2}}} & {- \frac{\left( {u \times a} \right)^{2}}{\sqrt{{a^{2}\left( {u \times a} \right)}^{2}}}} & 0 \\u & \frac{a}{\sqrt{a^{2}}} & {- \frac{\left( {{u^{0}a} - {a^{0}u}} \right) \times \left( {u \times a} \right)}{\sqrt{{a^{2}\left( {u \times a} \right)}^{2}}}} & \frac{u \times a}{\sqrt{\left( {u \times a} \right)^{2}}}\end{pmatrix}$ ${\overset{\circ}{L}\eta}:=\begin{pmatrix}u^{0} & \sqrt{u^{2}} & 0 & 0 \\u & {\frac{u^{0}}{\sqrt{u^{2}}}u} & {- \frac{u \times \left( {u \times a} \right)}{\sqrt{\left\lbrack {u \times \left( {u \times a} \right)} \right\rbrack^{2}}}} & \frac{u \times a}{\sqrt{\left( {u \times a} \right)^{2}}}\end{pmatrix}$

The rotation R̊η linking the two local frames is determined for a²≠0Λu²≠0in [12, equ.(61) and (62) on p. 24 and annex 20 on p. 85-87] using4×4-matrices as follows:

$\begin{matrix}{{\overset{\circ}{R}\eta} = {{\left( {\overset{\circ}{L}\eta} \right)^{- 1}\overset{\smile}{L}\eta} = {\begin{pmatrix}1 & 0^{T} \\0 & {\overset{\circ}{R}(\tau)}\end{pmatrix} = \begin{pmatrix}1 & 0 & 0 & 0 \\0 & \frac{a^{0}}{\sqrt{a^{2}}\sqrt{u^{2}}} & {- \frac{\sqrt{\left( {u \times a} \right)^{2}}}{\sqrt{a^{2}}\sqrt{u^{2}}}} & 0 \\0 & \frac{\sqrt{\left( {u \times a} \right)^{2}}}{\sqrt{a^{2}}\sqrt{u^{2}}} & \frac{a^{0}}{\sqrt{a^{2}}\sqrt{u^{2}}} & 0 \\0 & 0 & 0 & 1\end{pmatrix}}}} & (33)\end{matrix}$

This rotation matrix can easier be determined according to the inventionas defined in claim 1 by using 3×3 matrices and equ.(24):

$\begin{matrix}{{\overset{\circ}{R}\eta} = {\begin{pmatrix}1 & 0 \\0 & {\overset{\circ}{R}(\tau)}\end{pmatrix} = {{\left( {\overset{\circ}{L}\eta} \right)^{- 1}\overset{\smile}{L}\eta} = {\overset{\begin{matrix}{{equ}.{(24)}} \\ \downarrow \end{matrix}}{=}\text{ }{{\begin{pmatrix}1 & 0^{T} \\0 & {\overset{\circ}{L}{\,^{T}\left( {1 - {\frac{1}{\left( u^{0} \right)^{2}}{uu}^{T}}} \right)}\overset{\smile}{L}}\end{pmatrix}{\overset{\circ}{R}(\tau)}} = {{{\overset{\circ}{L}{\,^{T}\left( {1 - {\frac{1}{\left( u^{0} \right)^{2}}{uu}^{T}}} \right)}\overset{\smile}{L}} == \text{ }{\left( {{\frac{u^{0}}{\sqrt{u^{2}}}u} - {\frac{u \times \left( {u \times a} \right)}{\sqrt{\left\lbrack {u \times \left( {u \times a} \right)} \right\rbrack^{2}}}\frac{u \times a}{\sqrt{\left( {u \times a} \right)^{2}}}}} \right)^{T}{\left( {1 - {\frac{1}{\left( u^{0} \right)^{2}}{uu}^{T}}} \right) \cdot \cdot \text{ }\left( {\frac{a}{\sqrt{a^{2}}} - {\frac{\left( {{u^{0}a} - {a^{0}u}} \right) \times \left( {u \times a} \right)}{\sqrt{{a^{2}\left( {u \times a} \right)}^{2}}}\frac{u \times a}{\sqrt{\left( {u \times a} \right)^{2}}}}} \right)}}} =}}}}}} & (34)\end{matrix}$ $\begin{matrix}{\overset{\begin{matrix}{{annex}9} \\ \downarrow \end{matrix}}{=}\begin{pmatrix}\frac{a^{0}}{\sqrt{u^{2}}\sqrt{a^{2}}} & {- \frac{\sqrt{\left( {u \times a} \right)^{2}}}{\sqrt{u^{2}}\sqrt{a^{2}}}} & 0 \\\frac{\sqrt{\left( {u \times a} \right)^{2}}}{\sqrt{u^{2}}\sqrt{a^{2}}} & \frac{a^{0}}{\sqrt{u^{2}}\sqrt{a^{2}}} & 0 \\0 & 0 & 1\end{pmatrix}} & (35)\end{matrix}$

This is the result obtained in [12] as shown in equ.(33) above. Theremarks regarding the industrial applicability in annex 33 on p. 127 of[12], which is incorporated herein by reference, thus apply mutatismutandis also to this application.

Annex 1

The relationship

$\left( {1 + \frac{{uu}^{T}}{u^{0} + 1}} \right)^{- 1} = \left( {1 - \frac{{uu}^{T}}{u^{0}\left( {u^{0} + 1} \right)}} \right)$

is proven as follows:

${{\left( {1 + \frac{{uu}^{T}}{u^{0} + 1}} \right)\left( {1 - \frac{{uu}^{T}}{u^{0}\left( {u^{0} + 1} \right)}} \right)}=={1 + \frac{{uu}^{T}}{u^{0} + 1} - \frac{{uu}^{T}}{u^{0}\left( {u^{0} + 1} \right)} - {\left( \frac{{uu}^{T}}{u^{0} + 1} \right)\left( \frac{{uu}^{T}}{u^{0}\left( {u^{0} + 1} \right)} \right)}} == {1 + \frac{{uu}^{T}}{u^{0} + 1} - \frac{{uu}^{T}}{u^{0}\left( {u^{0} + 1} \right)} - \frac{u^{2}{uu}^{T}}{{u^{0}\left( {u^{0} + 1} \right)}^{2}}} == {1 + {\frac{{uu}^{T}}{u^{0} + 1}\left( {1 - \frac{1}{u^{0}} - \frac{u^{2}}{u^{0}\left( {u^{0} + 1} \right)}} \right)}} == {1 + {\frac{{uu}^{T}}{u^{0} + 1}\frac{{u^{0}\left( {u^{0} + 1} \right)} - \left( {u^{0} + 1} \right) - u^{2}}{u^{0}\left( {u^{0} + 1} \right)}}} == {1 + {\frac{{uu}^{T}}{u^{0} + 1}\frac{\left( u^{0} \right)^{2} - 1 - u^{2}}{u^{0}\left( {u^{0} + 1} \right)}}}} = {{1 + {\frac{{uu}^{T}}{u^{0} + 1}\frac{\left( u^{0} \right)^{2} - \left( u^{0} \right)^{2}}{u^{0}\left( {u^{0} + 1} \right)}}} = 1}$

Annex 2.1.

Since the four-velocity of a particle at rest is according to equ.(2)with v=0 equal to (1 0)^(T) and since a restricted Lorentztransformation must transform this to the four-velocity of equ.(2) withv=v, the first column vector of any restricted Lorentz transformationmatrix must be the four-velocity. One can thus make the followinggeneral ansatz for an arbitrary restricted Lorentz transformationmatrix:

${L\eta} = \begin{pmatrix}u^{0} & w^{T} \\u & L\end{pmatrix}$

From condition (6) above follows

$\left( {L\eta} \right)^{- 1} = {{{\eta\left( {L\eta} \right)}^{T}\eta} = {\begin{pmatrix}u^{0} & {- u^{T}} \\{- w} & L^{T}\end{pmatrix}}}$$1 = {{L{\eta\left( {L\eta} \right)}^{- 1}} = \begin{pmatrix}{\left( u^{0} \right)^{2} - w^{2}} & {- \left( {{u^{0}u} - {Lw}} \right)^{T}} \\{{u^{0}u} - {Lw}} & {{- {uu}^{T}} + {LL}^{T}}\end{pmatrix}}$ ⇓ $\begin{matrix}{{{1.\left( u^{0} \right)^{2}} - w^{2}} = {\left. 1\Leftrightarrow w^{2} \right. = u^{2}}} & (36)\end{matrix}$ $\begin{matrix}{{{2.u^{0}u} - {Lw}} = {{\left( {{u^{0}\hat{u}} - {L\hat{w}}} \right)\sqrt{u^{2}}} = {\left. 0\Leftrightarrow w \right. = {u^{0}L^{- 1}u}}}} & \text{(37)}\end{matrix}$ $\begin{matrix}{{3. - {uu}^{T} + {LL}^{T}} = {\left. 1\Leftrightarrow L^{- 1} \right. = {L^{T}\left( {1 + {uu}^{T}} \right)}^{- 1}}} & (38)\end{matrix}$ $\begin{matrix}{\left. \left\lbrack {6,{{fact}\left( {3.21\text{.1}} \right){on}p\text{.351}}} \right\rbrack\rightarrow\Leftrightarrow L^{- 1} \right. = {L^{T}\left( {1 - \frac{{uu}^{T}}{1 + u^{2}}} \right)}} & (39)\end{matrix}$ $\begin{matrix}{\left. \Leftrightarrow L^{- 1} \right. = {L^{T}\left( {1 - \frac{{uu}^{T}}{\left( u^{0} \right)^{2}}} \right)}} & (40)\end{matrix}$

From equations (37) and (39) follows

$w = {{u^{0}L^{- 1}u} = {{{u^{0}{L^{T}\left( {1 - \frac{{uu}^{T}}{1 + u^{2}}} \right)}u} == {u^{0}{L^{T}\left( {1 - \frac{u^{2}}{1 + u^{2}}} \right)}u}} = {{u^{0}L^{T}\frac{1}{\left( u^{0} \right)^{2}}u} = {\frac{1}{u^{0}}L^{T}u}}}}$

According to textbook [6, equ.(3.9.11) on p. 303 and fact 3.17.2 on p.334]

${\det\left( {L\eta} \right)} = {{\det\begin{pmatrix}u^{0} & w^{T} \\u & L\end{pmatrix}} = {{{{\det(L)}\left( {u^{0} - {w^{T}L^{- 1}u}} \right)}==\text{ }{{\det(L)}\left( {u^{0} - \frac{w^{T}u^{0}L^{- 1}u}{u^{0}}} \right)}}\overset{\begin{matrix}{{equ}.{(37)}} \\ \downarrow \end{matrix}}{=}{= {{{\det(L)}\left( {u^{0} - \frac{w^{T}w}{u^{0}}} \right)}\overset{\begin{matrix}{{equ}.{(36)}} \\ \downarrow \end{matrix}}{=}{\frac{\det(L)}{u^{0}}\overset{\begin{matrix}{{equ}.{(8)}} \\ \downarrow \end{matrix}}{=}{\left. 1\Rightarrow{\det(L)} \right. = u^{0}}}}}}}$

Annex 2.2.

Starting from the decomposition of equ.(13 one can derive

${L\eta} = {{\left( {B\eta} \right)\left( {R\eta} \right)} = {{\begin{pmatrix}u^{0} & u^{T} \\u & B\end{pmatrix}\begin{pmatrix}1 & 0^{T} \\0 & R\end{pmatrix}} = {\begin{pmatrix}u^{0} & {u^{T}R} \\u & {BR}\end{pmatrix} = \ldots}}}$

For the further transformation we need the following relationship:

$\left( {\frac{1}{u^{0}}({BR})^{T}u} \right)^{T} = {{\left( {\frac{1}{u^{0}}R^{T}B^{T}u} \right)^{T} == \text{ }\left( {\frac{1}{u^{0}}R^{T}\left( {1 + \frac{{uu}^{T}}{u^{0} + 1}} \right)^{T}u} \right)^{T}} = {{\left( {\frac{1}{u^{0}}{R^{T}\left( {1 + \frac{{uu}^{T}}{u^{0} + 1}} \right)}u} \right)^{T} == \text{ }\left( {\frac{1}{u^{0}}{R^{T}\left( {1 + \frac{u^{2}}{u^{0} + 1}} \right)}u} \right)^{T}} = {{{\left( {\frac{1}{u^{0}}\left( {1 + \frac{u^{2}}{u^{0} + 1}} \right)} \right)u^{T}R}==\text{ }{\frac{u^{0} + 1 + u^{2}}{u^{0}\left( {u^{0} + 1} \right)}u^{T}R}} = {{\left( \frac{u^{0} + \left( u^{0} \right)^{2}}{u^{0}\left( {u^{0} + 1} \right)} \right)u^{T}R} = {u^{T}R}}}}}$

Now we can resume the transformation:

$\ldots = {\begin{pmatrix}u^{0} & \left( {\frac{1}{u^{0}}({BR})^{T}u} \right)^{T} \\u & {BR}\end{pmatrix} = \begin{pmatrix}u^{0} & \left( {\frac{1}{u^{0}}L^{T}u} \right)^{T} \\u & L\end{pmatrix}}$

Further the following applies:

${LL}^{T} = {{{BR}({BR})}^{T} = {{{BRR}^{T}B^{T}} = {B^{2} = {{\left( {1 + \frac{{uu}^{T}}{u^{0} + 1}} \right)^{2} == \text{ }{1 + \frac{{uu}^{T}}{u^{0} + 1} + {\left( {1 + \frac{{uu}^{T}}{u^{0} + 1}} \right)\frac{{uu}^{T}}{u^{0} + 1}}} == {1 + \frac{{uu}^{T}}{u^{0} + 1} + {\left( {1 + \frac{u^{2}}{u^{0} + 1}} \right)\frac{{uu}^{T}}{u^{0} + 1}}} == {1 + {\left( {2 + \frac{u^{2}}{u^{0} + 1}} \right)\frac{{uu}^{T}}{u^{0} + 1}}}} = {{{1 + {\left( \frac{{2u^{0}} + 1 + 1 + u^{2}}{u^{0} + 1} \right)\frac{{uu}^{T}}{u^{0} + 1}}} == {1 + {\left( \frac{{2u^{0}} + 1 + \left( u^{0} \right)^{2}}{u^{0} + 1} \right)\frac{{uu}^{T}}{u^{0} + 1}}}} = {\left. {1 + {uu}^{T}}\Leftrightarrow L^{- 1} \right. = {L^{T}\left( {1 - \frac{{uu}^{T}}{\left( u^{0} \right)^{2}}} \right)}}}}}}}$and ${\det L} = {{\det({BR})} = {{\det B\det R}\overset{\begin{matrix}{{equ}.{(9)}} \\ \downarrow \end{matrix}}{=}{{{\det B} == {\det\left( {1 + \frac{{uu}^{T}}{u^{0} + 1}} \right)}}\overset{\begin{matrix}{\lbrack{6,{{fact}{({3.21\text{.1}})}{on}p\text{.351}}}\rbrack} \\ \downarrow \end{matrix}}{=}{{{1 + \frac{u^{2}}{u^{0} + 1}} == \frac{u^{0} + 1 + u^{2}}{u^{0} + 1}} = {\frac{u^{0} + \left( u^{0} \right)^{2}}{u^{0} + 1} = u^{0}}}}}}$

Annex 3

This annex details the transformation of equ.(26) into equ.(27)

$\begin{matrix}{{{equ}.(26)} = {\left\lbrack {1 - \frac{\left( {{u_{2}^{0}u_{1}} + {\left( {1 + \frac{u_{1}u_{1}^{T}}{u_{1}^{0} + 1}} \right)u_{2}}} \right) \cdot \left( {{u_{2}^{0}u_{1}} + {\left( {1 + \frac{u_{1}u_{1}^{T}}{u_{1}^{0} + 1}} \right)u_{2}}} \right)^{T}}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}} \right\rbrack \cdot \cdot}} & \text{(41)}\end{matrix}$$\left\lbrack {{u_{1}u_{2}^{T}} + {\left( {1 + \frac{u_{1}u_{1}^{T}}{u_{1}^{0} + 1}} \right)\left( {1 + \frac{u_{2}u_{2}^{T}}{u_{2}^{0} + 1}} \right)}} \right\rbrack ==$$\left\lbrack {1 - \frac{\left\lbrack {{\left( {u_{2}^{0} + \frac{u_{1}^{T}u_{2}}{u_{1}^{0} + 1}} \right)u_{1}} + u_{2}} \right\rbrack \cdot \left\lbrack {{\left( {u_{2}^{0} + \frac{u_{1}^{T}u_{2}}{u_{1}^{0} + 1}} \right)u_{1}} + u_{2}} \right\rbrack^{T}}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}} \right\rbrack \cdot \cdot$$\left\lbrack {1 + \frac{u_{1}u_{1}^{T}}{u_{1}^{0} + 1} + \frac{u_{2}u_{2}^{T}}{u_{2}^{0} + 1} + {\left( {1 + \frac{\left( {u_{1}^{T}u_{2}} \right)}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}} \right)u_{1}u_{2}^{T}}} \right\rbrack ==$$\left\lbrack {1 - {\frac{1}{\left( {u_{1}^{0} + 1} \right)^{2}} \cdot \cdot \frac{\left\lbrack {{\left( {u_{2}^{0} + {u_{2}^{0}u_{1}^{0}} + {u_{1}^{T}u_{2}}} \right)u_{1}} + {\left( {u_{1}^{0} + 1} \right)u_{2}}} \right\rbrack \cdot \text{ }\left\lbrack {{\left( {u_{2}^{0} + {u_{2}^{0}u_{1}^{0}} + {u_{1}^{T}u_{2}}} \right)u_{1}} + {\left( {u_{1}^{0} + 1} \right)u_{2}}} \right\rbrack^{T}}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}}} \right\rbrack \cdot \cdot$$\left\lbrack {1 + \frac{u_{1}u_{1}^{T}}{u_{1}^{0} + 1} + \frac{u_{2}u_{2}^{T}}{u_{2}^{0} + 1} + {\left( \frac{\begin{matrix}{{u_{1}^{0}u_{2}^{0}} + u_{1}^{0} +} \\{u_{2}^{0} + 1 + \left( {u_{1}^{T}u_{2}} \right)}\end{matrix}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} \right)u_{1}u_{2}^{T}}} \right\rbrack = \overset{\overset{u_{3}^{0} = {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}{({{equ}.{(15)}})}}}}{\downarrow}}{=}$$\left\lbrack {1 - {\frac{1}{\left( {u_{1}^{0} + 1} \right)^{2}} \cdot \cdot \frac{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}} + {\left( {u_{1}^{0} + 1} \right)u_{2}}} \right\rbrack \cdot \text{ }\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}} + {\left( {u_{1}^{0} + 1} \right)u_{2}}} \right\rbrack^{T}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack \cdot \cdot$$\left\lbrack {1 + \frac{u_{1}u_{1}^{T}}{u_{1}^{0} + 1} + \frac{u_{2}u_{2}^{T}}{u_{2}^{0} + 1} + {\frac{u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}u_{1}u_{2}^{T}}} \right\rbrack ==$$1 - {\frac{1}{\left( {u_{1}^{0} + 1} \right)^{2}}{\frac{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}} + {\left( {u_{1}^{0} + 1} \right)u_{2}}} \right\rbrack \cdot \left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}} + {\left( {u_{1}^{0} + 1} \right)u_{2}}} \right\rbrack^{T}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}++}}$$\left\lbrack {1 - {\frac{1}{\left( {u_{1}^{0} + 1} \right)^{2}}\frac{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}} + {\left( {u_{1}^{0} + 1} \right)u_{2}}} \right\rbrack \cdot \left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}} + {\left( {u_{1}^{0} + 1} \right)u_{2}}} \right\rbrack^{T}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack \cdot \cdot$${{\frac{u_{1}u_{1}^{T}}{u_{1}^{0} + 1}++}\left\lbrack {1 - {\frac{1}{\left( {u_{1}^{0} + 1} \right)^{2}}\frac{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}} + {\left( {u_{1}^{0} + 1} \right)u_{2}}} \right\rbrack \cdot \text{ }\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}} + {\left( {u_{1}^{0} + 1} \right)u_{2}}} \right\rbrack^{T}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack} \cdot \cdot$${{\frac{u_{2}u_{2}^{T}}{u_{2}^{0} + 1}++}\left\lbrack {1 - {\frac{1}{\left( {u_{1}^{0} + 1} \right)^{2}}\frac{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}} + {\left( {u_{1}^{0} + 1} \right)u_{2}}} \right\rbrack \cdot \text{ }\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}} + {\left( {u_{1}^{0} + 1} \right)u_{2}}} \right\rbrack^{T}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack} \cdot \cdot$${\frac{u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}u_{1}u_{2}^{T}} == {1 - {\frac{u_{2}^{0} + u_{3}^{0}}{\left( {u_{1}^{0} + 1} \right)^{2}}\frac{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{1}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{1}^{T}}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + -}$$\frac{1}{u_{1}^{0} + 1}{\frac{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}++}$$\left\lbrack {{u_{1}u_{1}^{T}} - {\frac{1}{\left( {u_{1}^{0} + 1} \right)^{2}} \cdot \cdot \frac{\begin{matrix}{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{1}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{1}^{T}}} \right\rbrack \cdot} \\\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}^{T}u_{1}} + {\left( {u_{1}^{0} + 1} \right)u_{2}^{T}u_{1}}} \right\rbrack\end{matrix}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack \cdot {\frac{1}{u_{1}^{0} + 1}++}$$\left\lbrack {{u_{2}u_{2}^{T}} - {\frac{1}{\left( {u_{1}^{0} + 1} \right)^{2}} \cdot \cdot \frac{\begin{matrix}{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}} \right\rbrack \cdot} \\\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}^{T}u_{2}} + {\left( {u_{1}^{0} + 1} \right)u_{2}^{T}u_{2}}} \right\rbrack\end{matrix}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack \cdot {\frac{1}{u_{2}^{0} + 1}++}$$\left\lbrack {{u_{1}u_{2}^{T}} - {\frac{1}{\left( {u_{1}^{0} + 1} \right)^{2}} \cdot \cdot \frac{\begin{matrix}{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}} \right\rbrack \cdot} \\\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}^{T}u_{1}} + {\left( {u_{1}^{0} + 1} \right)u_{2}^{T}u_{1}}} \right\rbrack\end{matrix}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack \cdot \cdot$$\frac{u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1}{\left( {u_{1}^{0} + 1} \right) + \left( {u_{2}^{0} + 1} \right)} == {1 - {\frac{u_{2}^{0} + u_{3}^{0}}{\left( {u_{1}^{0} + 1} \right)^{2}}\frac{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{1}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{1}^{T}}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + -}$$\frac{1}{u_{1}^{0} + 1}{\frac{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}++}$$\left\lbrack {{u_{1}u_{1}^{T}} + {- \frac{\begin{matrix}{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{1}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{1}^{T}}} \right\rbrack \cdot} \\\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)\left\lbrack {\left( u_{1}^{0} \right)^{2} - 1} \right\rbrack} + {\left( {u_{1}^{0} + 1} \right)u_{2}^{T}u_{1}}} \right\rbrack\end{matrix}}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}}} \right\rbrack \cdot \cdot {\frac{1}{u_{1}^{0} + 1}++}$$\left\lbrack {{u_{2}u_{2}^{T}} + {- \frac{\begin{matrix}{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}} \right\rbrack \cdot} \\\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}^{T}u_{2}} + {\left( {u_{1}^{0} + 1} \right)\left\lbrack {\left( u_{2}^{0} \right)^{2} - 1} \right\rbrack}} \right.\end{matrix}}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}}} \right\rbrack \cdot \cdot {\frac{1}{u_{2}^{0} + 1}++}$${\left\lbrack {{u_{1}u_{2}^{T}} + {- \frac{\begin{matrix}{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}} \right\rbrack \cdot} \\\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)\left\lbrack {\left( u_{1}^{0} \right)^{2} - 1} \right\rbrack} + {\left( {u_{1}^{0} + 1} \right)u_{2}^{T}u_{1}}} \right\rbrack\end{matrix}}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}}} \right\rbrack \cdot \cdot \frac{u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1}{\left( {u_{1}^{0} + 1} \right) + \left( {u_{2}^{0} + 1} \right)}} ==$$1 - {\frac{u_{2}^{0} + u_{3}^{0}}{\left( {u_{1}^{0} + 1} \right)^{2}}\frac{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{1}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{1}^{T}}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + -$$\frac{1}{u_{1}^{0} + 1}{\frac{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}++}$$\left\lbrack {{u_{1}u_{1}^{T}} - \frac{\begin{matrix}{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{1}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{1}^{T}}} \right\rbrack \cdot} \\\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)\left( {u_{1}^{0} - 1} \right)} + {u_{2}^{T}u_{1}}} \right\rbrack\end{matrix}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack \cdot {\frac{1}{u_{1}^{0} + 1}++}$$\left\lbrack {{u_{2}u_{2}^{T}} - {\begin{bmatrix}{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} +} \\{\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}\end{bmatrix} \cdot \cdot \frac{\begin{bmatrix}{{\left( {u_{2}^{0} + u_{3}^{0}} \right)\left( {{u_{1}^{T}u_{2}} + {u_{1}^{0}u_{2}^{0}} - {u_{1}^{0}u_{2}^{0}}} \right)} +} \\{\left( {u_{1}^{0} + 1} \right)\left\lbrack {\left( u_{2}^{0} \right)^{2} - 1} \right\rbrack}\end{bmatrix}}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}}} \right\rbrack \cdot$${{\frac{1}{u_{2}^{0} + 1}++}\left\lbrack {{u_{1}u_{2}^{T}} - \frac{\begin{matrix}{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}} \right\rbrack \cdot} \\\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)\left( {u_{1}^{0} - 1} \right)} + {u_{2}^{T}u_{1}}} \right\rbrack\end{matrix}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack} \cdot \cdot$$\frac{u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1}{\left( {u_{1}^{0} + 1} \right) + \left( {u_{2}^{0} + 1} \right)} == {1 - {\frac{u_{2}^{0} + u_{3}^{0}}{\left( {u_{1}^{0} + 1} \right)^{2}}\frac{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{1}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{1}^{T}}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + -}$$\frac{1}{u_{1}^{0} + 1}{\frac{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}++}$$\left\lbrack {{u_{1}u_{1}^{T}} - \frac{\begin{matrix}{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{1}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{1}^{T}}} \right\rbrack \cdot} \\\left\lbrack {{u_{2}^{0}u_{1}^{0}} + {u_{3}^{0}u_{1}^{0}} - u_{2}^{0} - u_{3}^{0} + {u_{2}^{T}u_{1}}} \right\rbrack\end{matrix}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack \cdot \cdot {\frac{1}{u_{1}^{0} + 1}++}$$\left\lbrack {{u_{2}u_{2}^{T}} - {\begin{bmatrix}{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} +} \\{\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}\end{bmatrix} \cdot \cdot \frac{\begin{bmatrix}{{\left( {u_{2}^{0} + u_{3}^{0}} \right)\left( {{u_{1}^{T}u_{2}} + {u_{1}^{0}u_{2}^{0}} - {u_{1}^{0}u_{2}^{0}}} \right)} +} \\{\left. \left( {u_{1}^{0} + 1} \right) \right)\left\lbrack {\left( u_{2}^{0} \right)^{2} - 1} \right\rbrack}\end{bmatrix}}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}}} \right\rbrack \cdot$${{\frac{1}{u_{2}^{0} + 1}++}\left\lbrack {{u_{1}u_{2}^{T}} - \frac{\begin{matrix}{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}} \right\rbrack \cdot} \\\left\lbrack {{u_{2}^{0}u_{1}^{0}} + {u_{3}^{0}u_{1}^{0}} - u_{2}^{0} - u_{3}^{0} + {u_{2}^{T}u_{1}}} \right\rbrack\end{matrix}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack} \cdot \cdot$$\frac{u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1}{\left( {u_{1}^{0} + 1} \right) + \left( {u_{2}^{0} + 1} \right)} == {1 - {\frac{u_{2}^{0} + u_{3}^{0}}{\left( {u_{1}^{0} + 1} \right)^{2}}\frac{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{1}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{1}^{T}}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + -}$$\frac{1}{u_{1}^{0} + 1}{\frac{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}++}$$\left\lbrack {{u_{1}u_{1}^{T}} - \frac{\begin{matrix}{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{1}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{1}^{T}}} \right\rbrack \cdot} \\\left\lbrack {u_{3}^{0} + {u_{3}^{0}u_{1}^{0}} - u_{2}^{0} - u_{3}^{0}} \right\rbrack\end{matrix}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack \cdot {\frac{1}{u_{1}^{0} + 1}++}$$\left\lbrack {{u_{2}u_{2}^{T}} - {\begin{bmatrix}{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} +} \\{\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}\end{bmatrix} \cdot \cdot \frac{\begin{bmatrix}{{\left( {u_{2}^{0} + u_{3}^{0}} \right)\left( {u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} +} \\{\left( {u_{1}^{0} + 1} \right)\left\lbrack {\left( u_{2}^{0} \right)^{2} - 1} \right\rbrack}\end{bmatrix}}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}}} \right\rbrack \cdot {\frac{1}{u_{2}^{0} + 1}++}$${\left\lbrack {{u_{1}u_{2}^{T}} - \frac{\begin{matrix}{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}} \right\rbrack \cdot} \\\left\lbrack {u_{3}^{0} + {u_{3}^{0}u_{1}^{0}} - u_{2}^{0} - u_{3}^{0}} \right\rbrack\end{matrix}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack \cdot \cdot \frac{u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1}{\left( {u_{1}^{0} + 1} \right) + \left( {u_{2}^{0} + 1} \right)}} ==$$1 - {\frac{u_{2}^{0} + u_{3}^{0}}{\left( {u_{1}^{0} + 1} \right)^{2}}\frac{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{1}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{1}^{T}}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + -$$\frac{1}{u_{1}^{0} + 1}{\frac{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}++}$$\left\lbrack {{u_{1}u_{1}^{T}} - \frac{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{1}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{1}^{T}}} \right\rbrack \cdot \left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack \cdot {\frac{1}{u_{1}^{0} + 1}++}$$\left\lbrack {{u_{2}u_{2}^{T}} - {\begin{bmatrix}{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} +} \\{\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}\end{bmatrix} \cdot \cdot \frac{\begin{bmatrix}{{u_{2}^{0}\left( {u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} + {u_{3}^{0}\left( {u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} +} \\{\left( {u_{1}^{0} + 1} \right)\left\lbrack {\left( u_{2}^{0} \right)^{2} - 1} \right\rbrack}\end{bmatrix}}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}}} \right\rbrack \cdot$${{\frac{1}{u_{2}^{0} + 1}++}\left\lbrack {{u_{1}u_{2}^{T}} - \frac{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}} \right\rbrack \cdot \left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack} \cdot \cdot$$\frac{u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1}{\left( {u_{1}^{0} + 1} \right) + \left( {u_{2}^{0} + 1} \right)} == {1 - {\frac{u_{2}^{0} + u_{3}^{0}}{\left( {u_{1}^{0} + 1} \right)^{2}}\frac{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{1}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{1}^{T}}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + -}$$\frac{1}{u_{1}^{0} + 1}{\frac{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}++}$$\left\lbrack {{u_{1}u_{1}^{T}} - \frac{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{1}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{1}^{T}}} \right\rbrack \cdot \left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack \cdot {\frac{1}{u_{1}^{0} + 1}++}$$\left\lbrack {{u_{2}u_{2}^{T}} - {\begin{bmatrix}{{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} +} \\{\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}\end{bmatrix} \cdot \cdot \frac{\left\lbrack {{u_{3}^{0}\left( {u_{2}^{0} + u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} + \left( u_{2}^{0} \right)^{2} - u_{1}^{0} - 1} \right\rbrack}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}}} \right\rbrack \cdot$${{\frac{1}{u_{2}^{0} + 1}++}\left\lbrack {{u_{1}u_{2}^{T}} - \frac{\left\lbrack {{\left( {u_{2}^{0} + u_{3}^{0}} \right)u_{1}u_{2}^{T}} + {\left( {u_{1}^{0} + 1} \right)u_{2}u_{2}^{T}}} \right\rbrack \cdot \left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack} \cdot \cdot$$\frac{u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1}{\left( {u_{1}^{0} + 1} \right) + \left( {u_{2}^{0} + 1} \right)} ==$${1++}u_{1}u_{1}^{T}{\left\{ {{- \frac{\left( {u_{2}^{0} + u_{3}^{0}} \right)^{2}}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} + \frac{1}{u_{1}^{0} + 1} - \frac{\left( {u_{2}^{0} + u_{3}^{0}} \right)\left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\}++}$$u_{2}u_{2}^{T}{\begin{Bmatrix}{{- \frac{1}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + \frac{1}{u_{2}^{0} + 1} - \frac{{u_{3}^{0}\left( {u_{2}^{0} + u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} + \left( u_{2}^{0} \right)^{2} - u_{1}^{0} - 1}{\left( {u_{2}^{0} + 1} \right)\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + -} \\{\frac{\left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\frac{\left( {u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1} \right)}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}}\end{Bmatrix}++}$ $u_{1}u_{2}^{T}{\begin{Bmatrix}{{- \frac{u_{2}^{0} + u_{3}^{0}}{{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{1}^{0} + 1} \right)}} - {\frac{\begin{matrix}{\left( {u_{2}^{0} + u_{3}^{0}} \right) \cdot \left\lbrack {{u_{3}^{0}\left( {u_{2}^{0} + u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} +} \right.} \\\left. {\left( u_{2}^{0} \right)^{2} - u_{1}^{0} - 1} \right\rbrack\end{matrix}}{\left( {u_{2}^{0} + 1} \right)\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}++}} \\{\frac{u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}\left\lbrack {1 - \frac{\left( {u_{2}^{0} + u_{3}^{0}} \right) \cdot \left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack}\end{Bmatrix}++}$${u_{2}u_{1}^{T}\left\{ {{{- \frac{u_{2}^{0} + u_{3}^{0}}{\left( {u_{1}^{0} + 1} \right)^{2}}}\frac{u_{1}^{0} + 1}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} - \frac{\left( {u_{1}^{0} + 1} \right)\left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right.}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\}} =$${\overset{\overset{{annex}4}{\downarrow}}{=}{{1++}u_{1}u_{1}^{T}{\frac{1 - u_{2}^{0}}{\left( {u_{1}^{0} + 1} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}++}}}\text{ }{u_{2}u_{2}^{T}{\frac{1 - u_{1}^{0}}{\left( {u_{2}^{0} + 1} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}++}}\text{ }{{{u_{1}u_{2}^{T}\frac{{2u_{1}^{T}u_{2}} + {\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}} + {{- u_{2}}u_{1}^{T}\frac{1}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1}}} = {{equ}.(27)}}$

Annex 4

In this annex the scalar factor associated with each dyad u₁u₁ ^(T),u₂u₂ ^(T), u₁u₂ ^(T), u₂u₁ ^(T) in equ.(41) is simplified to getequ.(27):

$u_{1}{u_{1}^{T}:{\left\{ {{- \frac{\left( {u_{2}^{0} + u_{3}^{0}} \right)^{2}}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} + \frac{1}{u_{1}^{0} + 1} - \frac{\left( {u_{2}^{0} + u_{3}^{0}} \right)\left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\} ==}}$$\frac{\begin{matrix}{{- \left( {u_{2}^{0} + u_{3}^{0}} \right)^{2}} + {\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} -} \\{\left( {u_{2}^{0} + u_{3}^{0}} \right)\left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}\end{matrix}}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} == \frac{\begin{matrix}{{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} -} \\{\left( {u_{2}^{0} + u_{3}^{0}} \right)\left( {{u_{3}^{0}u_{1}^{0}} + u_{3}^{0}} \right)}\end{matrix}}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} ==$$\frac{{\left( {u_{1}^{0} + 1} \right)\left( {u_{3}^{0} + 1} \right)} - {\left( {u_{2}^{0} + u_{3}^{0}} \right)\left( {u_{1}^{0} + 1} \right)}}{\left( {u_{1}^{0} + 1} \right)^{2}\left( {u_{3}^{0} + 1} \right)} = {\frac{\left( {u_{3}^{0} + 1} \right) - \left( {u_{2}^{0} + u_{3}^{0}} \right)}{\left( {u_{1}^{0} + 1} \right)\left( {u_{3}^{0} + 1} \right)} ==}$$\frac{1 - u_{2}^{0}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{3}^{0} + 1} \right)} = {\overset{\overset{u_{3}^{0} = {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}{({{equ}.{(15)}})}}}}{\downarrow}}{=}\frac{1 - u_{2}^{0}}{\left( {u_{1}^{0} + 1} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}}$$u_{2}{u_{2}^{T}:\left\{ {{- \frac{1}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + \frac{1}{u_{2}^{0} + 1} - \frac{{u_{3}^{0}\left( {u_{2}^{0} + u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} + \left( u_{2}^{0} \right)^{2} - u_{1}^{0} - 1}{\left( {u_{2}^{0} + 1} \right)\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + -} \right.}$$\left. {\frac{\left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\frac{\left( {u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1} \right)}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}} \right\} ==$$\frac{\begin{matrix}{{{- \left( {u_{2}^{0} + 1} \right)}\left( {u_{1}^{0} + 1} \right)} + {\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} -} \\{{u_{3}^{0}\left( {u_{2}^{0} + u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} - \left( u_{2}^{0} \right)^{2} + u_{1}^{0} + 1}\end{matrix}}{\left( {u_{2}^{0} + 1} \right)\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + -$${\frac{\left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\frac{\left( {u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1} \right)}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}} ==$${\frac{\begin{matrix}{{\left( {u_{1}^{0} + 1} \right)\left\lbrack {{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)} - \left( {u_{2}^{0} + 1} \right) + 1} \right\rbrack} -} \\{{u_{3}^{0}\left( {u_{2}^{0} + u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} - \left( u_{2}^{0} \right)^{2}}\end{matrix}}{\left( {u_{2}^{0} + 1} \right)\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + {{- \frac{\left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}\frac{\left( {u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1} \right)}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}}} ==$${\frac{\begin{matrix}{{\left( {u_{1}^{0} + 1} \right)\left\lbrack {{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)} - u_{2}^{0}} \right\rbrack} -} \\{{u_{3}^{0}\left( {u_{2}^{0} + u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} - \left( u_{2}^{0} \right)^{2}}\end{matrix}}{\left( {u_{2}^{0} + 1} \right)\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + {{- \frac{\left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}\frac{\left( {u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1} \right)}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}}} ==$${\frac{\begin{matrix}{{u_{1}^{0}\left\lbrack {{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)} - u_{2}^{0}} \right\rbrack} + {u_{3}^{0}\left( {u_{3}^{0} + 1} \right)} -} \\{u_{2}^{0} - {u_{3}^{0}\left( {u_{2}^{0} + u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} - \left( u_{2}^{0} \right)^{2}}\end{matrix}}{\left( {u_{2}^{0} + 1} \right)\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + {{- \frac{\left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}\frac{\left( {u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1} \right)}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}}} ==$${\frac{\begin{matrix}{{u_{1}^{0}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} - {u_{1}^{0}u_{2}^{0}} + u_{3}^{0} -} \\{u_{2}^{0} - {u_{3}^{0}\left( {u_{2}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} - \left( u_{2}^{0} \right)^{2}}\end{matrix}}{\left( {u_{2}^{0} + 1} \right)\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + {{- \frac{\left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}\frac{\left( {u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1} \right)}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}}} ==$${\frac{\begin{matrix}{{u_{1}^{0}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} - {u_{1}^{0}u_{2}^{0}} + u_{3}^{0} -} \\{u_{2}^{0} - {u_{2}^{0}u_{3}^{0}} + {u_{1}^{0}u_{2}^{0}u_{3}^{0}} - \left( u_{2}^{0} \right)^{2}}\end{matrix}}{\left( {u_{2}^{0} + 1} \right)\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + {{- \frac{\left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}\frac{\left( {u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1} \right)}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}}} ==$${\frac{{u_{1}^{0}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + u_{3}^{0} + {u_{1}^{0}u_{2}^{0}u_{3}^{0}}}{\left( {u_{2}^{0} + 1} \right)\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} - {\frac{u_{3}^{0}u_{1}^{0}}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\frac{\left( {u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1} \right)}{\left. {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}}} ==$$\frac{u_{3}^{0}\left\lbrack {1 - \left( u_{1}^{0} \right)^{2}} \right\rbrack}{\left( {u_{2}^{0} + 1} \right)\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} =$$\frac{1 - u_{1}^{0}}{\left( {u_{2}^{0} + 1} \right)\left( {u_{3}^{0} + 1} \right)} = {\overset{\overset{u_{3}^{0} = {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}{({{equ}.{(15)}})}}}}{\downarrow}}{=}\frac{1 - u_{1}^{0}}{\left( {u_{2}^{0} + 1} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}}$$u_{1}{u_{2}^{T}:\left\{ {{- \frac{u_{2}^{0} + u_{3}^{0}}{{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{1}^{0} + 1} \right)}} -} \right.}$$\frac{\left( {u_{2}^{0} + u_{3}^{0}} \right) \cdot \left\lbrack {{u_{3}^{0}\left( {u_{2}^{0} + u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} + \left( u_{2}^{0} \right)^{2} - u_{1}^{0} - 1} \right\rbrack}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{2}^{0} + 1} \right)}++$$\left. {\frac{u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}\left\lbrack {1 - \frac{\left( {u_{2}^{0} + u_{3}^{0}} \right) \cdot \left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\rbrack} \right\} ==$${- \frac{u_{2}^{0} + u_{3}^{0}}{{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{1}^{0} + 1} \right)}} - {\frac{\left( {u_{2}^{0} + u_{3}^{0}} \right) \cdot \left\lbrack {{u_{3}^{0}\left( {u_{2}^{0} + u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} + \left( u_{2}^{0} \right)^{2} - u_{1}^{0} - 1} \right\rbrack}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{2}^{0} + 1} \right)}++}$${\frac{u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} \cdot \frac{{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} - {\left( {u_{2}^{0} + u_{3}^{0}} \right) \cdot \left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} ==$${- \frac{u_{2}^{0} + u_{3}^{0}}{{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{1}^{0} + 1} \right)}} - {\frac{\left( {u_{2}^{0} + u_{3}^{0}} \right) \cdot \left\lbrack {{u_{3}^{0}\left( {u_{2}^{0} + u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} + \left( u_{2}^{0} \right)^{2} - u_{1}^{0} - 1} \right\rbrack}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{2}^{0} + 1} \right)}++}$${\frac{u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} \cdot \frac{\begin{matrix}{{u_{1}^{0}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} + {u_{3}^{0}\left( {u_{3}^{0} + 1} \right)} - {u_{2}^{0}\left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)} -} \\{u_{3}^{0}\left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}\end{matrix}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} ==$${- \frac{u_{2}^{0} + u_{3}^{0}}{{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{1}^{0} + 1} \right)}} - {\frac{\left( {u_{2}^{0} + u_{3}^{0}} \right) \cdot \left\lbrack {{u_{3}^{0}\left( {u_{2}^{0} + u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} + \left( u_{2}^{0} \right)^{2} - u_{1}^{0} - 1} \right\rbrack}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{2}^{0} + 1} \right)}++}$${\frac{u_{3}^{0} + u_{1}^{0} + u_{2}^{0} + 1}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} \cdot \frac{{\left( {u_{1}^{0} + 1} \right)u_{3}^{0}} + {u_{3}^{0}\left( {u_{3}^{0} + u_{2}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} + \left( u_{2}^{0} \right)^{2}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} ==$${- \frac{u_{2}^{0} + u_{3}^{0}}{{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{1}^{0} + 1} \right)}} - {\frac{\left( {u_{2}^{0} + u_{3}^{0}} \right) \cdot \left\lbrack {{u_{3}^{0}\left( {u_{2}^{0} + u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} + \left( u_{2}^{0} \right)^{2} - u_{1}^{0} - 1} \right\rbrack}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{2}^{0} + 1} \right)}++}$$\frac{u_{3}^{0} + u_{2}^{0}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} \cdot {\frac{{\left( {u_{1}^{0} + 1} \right)u_{3}^{0}} + {u_{3}^{0}\left( {u_{3}^{0} + u_{2}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} + \left( u_{2}^{0} \right)^{2}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}++}$${\frac{u_{1}^{0} + 1}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} \cdot \frac{{\left( {u_{1}^{0} + 1} \right)u_{3}^{0}} + {u_{3}^{0}\left( {u_{3}^{0} + u_{2}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} + \left( u_{2}^{0} \right)^{2}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} ==$${- \frac{u_{2}^{0} + u_{3}^{0}}{{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{1}^{0} + 1} \right)}} + {{\frac{\left( {u_{2}^{0} + u_{3}^{0}} \right) \cdot \left( {u_{1}^{0} + 1} \right)}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{2}^{0} + 1} \right)}++}{\frac{u_{3}^{0} + u_{2}^{0}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} \cdot}}$${\frac{\left( {u_{1}^{0} + 1} \right)u_{3}^{0}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}++}{\frac{u_{1}^{0} + 1}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} \cdot}$$\frac{{\left( {u_{1}^{0} + 1} \right)u_{3}^{0}} + {u_{3}^{0}\left( {u_{3}^{0} + u_{2}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} + \left( u_{2}^{0} \right)^{2}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} == {{- \frac{\left( {u_{2}^{0} + u_{3}^{0}} \right)\left( {u_{2}^{0} + 1} \right)}{{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}} +}$${\frac{u_{2}^{0} + u_{3}^{0}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{2}^{0} + 1} \right)}++}{\frac{\left( {u_{3}^{0} + u_{2}^{0}} \right)u_{3}^{0}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{2}^{0} + 1} \right)}++}$$\frac{{\left( {u_{1}^{0} + 1} \right)u_{3}^{0}} + {u_{3}^{0}\left( {u_{3}^{0} + u_{2}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)} + \left( u_{2}^{0} \right)^{2}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{2}^{0} + 1} \right)} ==$$\frac{\left( u_{3}^{0} \right)^{2} + {\left( {u_{1}^{0} + 1} \right)u_{3}^{0}} + {u_{3}^{0}\left( {u_{3}^{0} + u_{2}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}\left( {u_{2}^{0} + 1} \right)} ==$$\frac{{2u_{3}^{0}} + u_{1}^{0} + u_{2}^{0} - {u_{1}^{0}u_{2}^{0}} + 1}{\left( {u_{1}^{0} + 1} \right)\left( {u_{3}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} = \overset{\overset{u_{3}^{0} = {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}{({{equ}.{(15)}})}}}}{\downarrow}}{=}$$\frac{{2\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}} \right)} + u_{1}^{0} + u_{2}^{0} - {u_{1}^{0}u_{2}^{0}} + 1}{\left( {u_{1}^{0} + 1} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{2}^{0} + 1} \right)} ==$$\frac{{2u_{1}^{T}u_{2}} + u_{1}^{0} + u_{2}^{0} + {u_{1}^{0}u_{2}^{0}} + 1}{\left( {u_{1}^{0} + 1} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{2}^{0} + 1} \right)} == \frac{{2u_{1}^{T}u_{2}} + {\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}$$u_{2}{u_{1}^{T}:{\left\{ {{{- \frac{u_{2}^{0} + u_{3}^{0}}{\left( {u_{1}^{0} +} \right)^{2}}}\frac{u_{1}^{0} + 1}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}} - \frac{\left( {u_{1}^{0} + 1} \right)\left( {{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}} \right)}{\left( {u_{1}^{0} + 1} \right)^{2}{u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} \right\} ==}}$${{{- \frac{u_{2}^{0} + u_{3}^{0}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} - \frac{{u_{3}^{0}u_{1}^{0}} - u_{2}^{0}}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}} == {- \frac{u_{3}^{0}\left( {u_{1}^{0} + 1} \right)}{\left( {u_{1}^{0} + 1} \right){u_{3}^{0}\left( {u_{3}^{0} + 1} \right)}}}} =$${- \frac{1}{u_{3}^{0} + 1}} = {\overset{\overset{u_{3}^{0} = {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}{({{equ}.{(15)}})}}}}{\downarrow}}{=}{- \frac{1}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1}}}$

Annex 5

In this annex equ. (27) is separated into the symmetric and theantisymmetric (skewsymmetric) part to get the Rodrigues form ofequ.(28):

${{equ}.(27)} = {1 + {u_{1}u_{1}^{T}{\frac{1 - u_{2}^{0}}{\left( {u_{1}^{0} + 1} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}++}}}$${u_{2}u_{2}^{T}{\frac{1 - u_{1}^{0}}{\left( {u_{2}^{0} + 1} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}++}u_{1}u_{2}^{T}\frac{{2u_{1}^{T}u_{2}} + {\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)}} +$${{- u_{2}}u_{1}^{T}\frac{1}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1}}=={1 + {\frac{1}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \cdot \cdot}}$$\left\lbrack {{u_{1}u_{1}^{T}\frac{1 - u_{2}^{0}}{u_{1}^{0} + 1}} + {u_{2}u_{2}^{T}\frac{1 - u_{1}^{0}}{u_{2}^{0} + 1}} + {u_{1}{u_{2}^{T}\left( {\frac{2u_{1}^{T}u_{2}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} + 1} \right)}} - {u_{2}u_{1}^{T}}} \right\rbrack =$$R = {{{\frac{1}{2}\left( {R + R^{T}} \right)} + {\frac{1}{2}\left( {R - R^{T}} \right)}}\overset{\downarrow}{=}{1 + {\frac{1}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \cdot \cdot}}}$$\left\lbrack {{u_{1}u_{1}^{T}\frac{1 - u_{2}^{0}}{u_{1}^{0} + 1}} + {u_{2}u_{2}^{T}{\frac{1 - u_{1}^{0}}{u_{2}^{0} + 1}++}\frac{1}{2}\left( {{u_{1}u_{2}^{T}} + {u_{2}u_{1}^{T}}} \right)\left( {\frac{2u_{1}^{T}u_{2}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} + 1} \right)} -} \right.$${\frac{1}{2}{\left( {{u_{2}u_{1}^{T}} + {u_{1}u_{2}^{T}}} \right)++}\frac{1}{2}\left( {{u_{1}u_{2}^{T}} - {u_{2}u_{1}^{T}}} \right)\left( {\frac{2u_{1}^{T}u_{2}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} + 1} \right)} -$$\left. {\frac{1}{2}\left( {{u_{2}u_{1}^{T}} - {u_{1}u_{2}^{T}}} \right)} \right\rbrack=={1 + {\frac{1}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1}\left\lbrack {{u_{1}u_{1}^{T}\frac{1 - u_{2}^{0}}{u_{1}^{0} + 1}} + {u_{2}u_{2}^{T}{\frac{1 - u_{1}^{0}}{u_{2}^{0} + 1}++}}} \right.}}$$\frac{1}{2}\left( {{u_{1}u_{2}^{T}} + {u_{2}u_{1}^{T}}} \right){\frac{2u_{1}^{T}u_{2}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}++}$$\left. {\frac{1}{2}\left( {{u_{1}u_{2}^{T}} - {u_{2}u_{1}^{T}}} \right)\left( {\frac{2u_{1}^{T}u_{2}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} + 2} \right)} \right\rbrack==$$1 + {\frac{1}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1}\left\lbrack {{u_{1}u_{1}^{T}\frac{1 - u_{2}^{0}}{u_{1}^{0} + 1}} + {u_{2}u_{2}^{T}{\frac{1 - u_{1}^{0}}{u_{2}^{0} + 1}++}}} \right.}$$\left. {{\left( {{u_{1}u_{2}^{T}} + {u_{2}u_{1}^{T}}} \right)\frac{u_{1}^{T}u_{2}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}} + {\left( {{u_{1}u_{2}^{T}} - {u_{2}u_{1}^{T}}} \right)\left( {\frac{u_{1}^{T}u_{2}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} + 1} \right)}} \right\rbrack==$$1 + {\frac{1}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} \cdot \left\lbrack {{u_{1}{u_{1}^{T}\left( {1 - u_{2}^{0}} \right)}\left( {u_{2}^{0} + 1} \right)} +} \right.}$u₂u₂^(T)(1 − u₁⁰)(u₁⁰ + 1) + +(u₁u₂^(T) + u₂u₁^(T))u₁^(T)u₂ + +(u₁u₂^(T) − u₂u₁^(T))[u₁^(T)u₂ + (u₁⁰ + 1)(u₂⁰ + 1)]] =  = 1+$\frac{1}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} \cdot \left\lbrack {{u_{1}{u_{1}^{T}\left\lbrack {1 - \left( u_{2}^{0} \right)^{2}} \right\rbrack}} + {u_{2}{{u_{2}^{T}\left\lbrack {1 - \left( u_{1}^{0} \right)^{2}} \right\rbrack}++}}} \right.$(u₁u₂^(T) + u₂u₁^(T))u₁^(T)u₂ + +(u₁u₂^(T) − u₂u₁^(T))[u₁^(T)u₂ + (u₁⁰ + 1)(u₂⁰ + 1)]]=$1 + {\frac{1}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} \cdot \left\lbrack {{{- \left( u_{2} \right)^{2}}u_{1}u_{1}^{T}} - {\left( u_{1} \right)^{2}u_{2}u_{2}^{T}} +} \right.}$(u₁u₂^(T) + u₂u₁^(T))u₁^(T)u₂ + −(u₂u₁^(T) − u₁u₂^(T))[u₁^(T)u₂ + (u₁⁰ + 1)(u₂⁰ + 1)]] = …

For the further transformation we need the following identities:

$\left\lbrack {u_{1} \times u_{2}} \right\rbrack_{\times}\overset{\overset{{{{\lbrack{6,{{fact}4.12\text{.1}{xi}}}})}{on}p\text{.384}}\rbrack}{\downarrow}}{=}\left. {{u_{2}u_{1}^{T}} - {u_{1}u_{2}^{T}}}\Rightarrow{\left\lbrack {u_{1} \times u_{2}} \right\rbrack_{\times}^{2}=={\left( {{u_{2}u_{1}^{T}} - {u_{1}u_{2}^{T}}} \right)\left( {{u_{2}u_{1}^{T}} - {u_{1}u_{2}^{T}}} \right)}=={{u_{2}u_{1}^{T}u_{2}u_{1}^{T}} - {u_{2}u_{1}^{T}u_{1}u_{2}^{T}} - {u_{1}u_{2}^{T}u_{2}u_{1}^{T}} + {u_{1}u_{2}^{T}u_{1}u_{2}^{T}}}=={{\left( {u_{1}^{T}u_{2}} \right)u_{2}u_{1}^{T}} - {\left( u_{1}^{2} \right)u_{2}u_{2}^{T}} - {\left( u_{2}^{2} \right)u_{1}u_{1}^{T}} + {\left( {u_{2}^{T}u_{1}} \right)u_{1}u_{2}^{T}}}=={{{- \left( u_{1}^{2} \right)}u_{2}u_{2}^{T}} + {\left( {u_{1}^{T}u_{2}} \right)\left( {{u_{2}u_{1}^{T}} + {u_{1}u_{2}^{T}}} \right)} - {\left( u_{2}^{2} \right)u_{1}u_{1}^{T}}}} \right.$

Now we can resume the transformation:

$\ldots = {1 + {\frac{1}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} \cdot}}$[[u₁ × u₂]_(×)² − [u₁ × u₂]_(×)[u₁^(T)u₂ + (u₁⁰ + 1)(u₂⁰+)]] = =$\underset{\frac{1}{2}{({R + R^{T}})}}{\underset{︸}{1 + {\frac{1}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}\left\lbrack {u_{1} \times u_{2}} \right\rbrack}_{\times}^{2}}} +$$\underset{\frac{1}{2}{({R - R^{T}})}}{\underset{︸}{- {\frac{{u_{1}^{T}u_{2}} + {\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}\left\lbrack {u_{1} \times u_{2}} \right\rbrack}_{\times}}} = \overset{\overset{{equ}.{(15)}}{\downarrow}}{=}$${1 - {{{\frac{u_{1}^{0} + u_{2}^{0} + \overset{u_{3}^{0}}{\overset{︷}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}}} + 1}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)\left( {\underset{u_{3}^{0}}{\underset{︸}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}}} + 1} \right)}\left\lbrack {u_{1} \times u_{2}} \right\rbrack}_{\times}++}{\frac{1}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)\left( {\underset{u_{3}^{0}}{\underset{︸}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}}} + 1} \right)}\left\lbrack {u_{1} \times u_{2}} \right\rbrack}_{\times}^{2}}} = {{equ}.(28)}$

Annex 6

In this annex we show that equ.(27) corresponds to the result obtainedby Fahnline [1, equ.(30) on p. 820]:

We denoted above the components of the four dimensional matrix ^(W)Rηrepresenting the mixed Wigner rotation tensor with R^(μ) _(ν) (μ, ν=0,1, 2, 3), the components of the three dimensional submatrix ^(W)R arethus ^(W)R^(i) _(j)=^(W)R^(i) _(j) with i, j=1, 2, 3. Comparing [1,equ.(21) on p. 820] with equ.(16) above one can see that Fahnlinedenotes the components of the three dimensional transposed submatrix^(W)R^(T) in equation [1, equ.(30) on p. 820] as R^(i) _(j), saidequation reads (there is a typing error at the end of the first line ofsaid equation: υ_(B) ^(i)b_(B) _(j) should read υ_(B) ^(i)υ_(B) _(j) ):

$\begin{matrix}{R_{j}^{i} = {\delta_{j}^{i} + {\left\lbrack {{\left( {1 - \gamma_{B}^{2}} \right)v_{A}^{i}v_{Aj}} + {\left( {1 - \gamma_{A}^{2}} \right)v_{B}^{i}v_{Bj}} + {{- \left( {1 + \gamma_{A}} \right)}\left( {1 + \gamma_{B}} \right)v_{A}^{i}{v_{Bj}++}\left( {1 + \gamma_{A} + \gamma_{B} + {3\gamma_{A}\gamma_{B}} - {2\psi}} \right)v_{B}^{i}v_{Aj}}} \right\rbrack \cdot \cdot \frac{1}{{c^{2}\left( {1 + \gamma_{A}} \right)}\left( {1 + \gamma_{B}} \right)\left( {1 + {2\gamma_{A}\gamma_{B}} - \psi} \right)}}}} & (42)\end{matrix}$

As already mentioned in the paragraph below equ.(3) above Fahnline [1]defines the four-velocity without the factor 1/c, therefore thefollowing identities apply:

$\frac{v_{A}^{i}v_{Aj}}{c^{2}}$

are the components of the dyad we denote with u₁u₁ ^(T),

$\frac{v_{B}^{i}v_{Bj}}{c^{2}}$

are the components of the dyad we denote with u₂u₂ ^(T),

$\frac{v_{A}^{i}v_{Bj}}{c^{2}}$

are the components of the dyad we denote with u₁u₂ ^(T), and

$\frac{v_{B}^{i}v_{Aj}}{c^{2}}$

are the components of the dyad we denote with u₂u₁ ^(T).

According to equ.(2) above and [1, paragraph between equations (25) and(26) on p. 820] further the following identities apply:

γ_(A) =u ₁ ⁰Λγ_(B) =u ₂ ⁰ Λψ=−u ₁ ·u ₂ =u ₁ ⁰ u ₂ ⁰ −u ₁ ^(T) u ₂

Thus equ.(42) translated into our notation and using matrix notationinstead of component notation reads

W R T = 1 + { [ 1 - ( u 2 0 ) 2 ] ⁢ u 1 ⁢ u 1 T + [ 1 - ( u 1 0 ) 2 ] ⁢ u 2⁢u 2 T + - ( 1 + u 1 0 ) ⁢ ( 1 + u 2 0 ) ⁢ u 1 ⁢ u 2 T ++ [ 1 + u 1 0 + u 20 + 3 ⁢ u 1 0 ⁢ u 2 0 - 2 ⁢ ( u 1 0 ⁢ u 2 0 - u 1 T ⁢ u 2 ) ] ⁢ u 2 ⁢ u 1 T } ·· 1 ( 1 + u 1 0 ) ⁢ ( 1 + u 2 0 ) [ 1 + 2 ⁢ u 1 0 ⁢ u 2 0 - ( u 1 0 ⁢ u 20 - u 1 T ⁢ u 2 ) ] == 1 + { ( 1 - u 2 0 ) ⁢ ( 1 + u 2 0 ) ⁢ u 1 ⁢ u 1 T + (1 - u 1 0 ) ⁢ ( 1 + u 1 0 ) ⁢ u 2 ⁢ u 2 T + - ( 1 + u 1 0 ) ⁢ ( 1 + u 2 0 ) ⁢u 1 ⁢ u 2 T ++ ⁢ ( 1 + u 1 0 + u 2 0 + u 1 0 ⁢ u 2 0 + 2 ⁢ u 1 T ⁢ u 2 ) ⁢ u 2⁢u 1 T } · · 1 ( 1 + u 1 0 ) ⁢ ( 1 + u 2 0 ) ⁢ ( 1 + u 1 0 ⁢ u 2 0 + u 1 T ⁢u 2 ) == 1 + 1 - u 2 0 ( 1 + u 1 0 ) ⁢ ( 1 + u 1 0 ⁢ u 2 0 + u 1 T ⁢ u 2 ) ⁢u 1 ⁢ u 1 T + 1 - u 1 0 ( 1 + u 2 0 ) ⁢ ( 1 + u 1 0 ⁢ u 2 0 + u 1 T ⁢ u 2 ) ⁢u 2 ⁢ u 2 T + - 1 1 + u 1 0 ⁢ u 2 0 + u 1 T ⁢ u 2 ⁢ u 1 ⁢ u 2 T ++ ⁢ 1 + u 10 + u 2 0 + u 1 0 ⁢ u 2 0 + 2 ⁢ u 1 T ⁢ u 2 ( 1 + u 1 0 ) ⁢ ( 1 + u 2 0 ) ⁢ (1 + u 1 0 ⁢ u 2 0 + u 1 T ⁢ u 2 ) ⁢ u 2 ⁢ u 1 T

This is the transposed of equ.(27) above.

Annex 7

In this annex the equivalence of equ.(31) and (32) is proven:

${{equ}.(31)} = {{{1 - \frac{\left( {u_{1} \times u_{2}} \right)^{2}}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}}=={1 - \frac{{u_{1}^{2}u_{2}^{2}} - \left( {u_{1}^{T}u_{2}} \right)^{2}}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}}=={1 - \frac{{u_{1}^{2}u_{2}^{2}} - \left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} - {u_{1}^{0}u_{2}^{0}}} \right)^{2}}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}}=={1 - \frac{{u_{1}^{2}u_{2}^{2}} - \left( {u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)^{2}}{\left( {u_{3}^{0} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}}=={1 - {\frac{1}{\left( {u_{3}^{0} + 1} \right)}\left\lbrack {\frac{\left( {u_{1}^{2} + 1 - 1} \right)\left( {u_{2}^{2} + 1 - 1} \right)}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} - \frac{\left( {u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)^{2}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}} \right\rbrack}}=={1 - {\frac{1}{\left( {u_{3}^{0} + 1} \right)}\left\lbrack {\frac{\left( {\left( u_{1}^{0} \right)^{2} - 1} \right)\left( {\left( u_{2}^{0} \right)^{2} - 1} \right)}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} - \frac{\left( {u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)^{2}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}} \right\rbrack}}=={1 - {\frac{1}{\left( {u_{3}^{0} + 1} \right)}\left\lbrack {{\left( {u_{1}^{0} - 1} \right)\left( {u_{2}^{0} - 1} \right)} - \frac{\left( {u_{3}^{0} - {u_{1}^{0}u_{2}^{0}}} \right)^{2}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}} \right\rbrack}}} = {\overset{\overset{\lbrack{4,{p\text{.213}},{{last}{two}{equations}}}\rbrack}{\downarrow}}{=}{{\frac{\left( {u_{1}^{0} + u_{2}^{0} + \overset{u_{3}^{0}}{\overset{︷}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}}} + 1} \right)^{2}}{\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)\left( {\underset{u_{3}^{0}}{\underset{︸}{{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}}}} + 1} \right)} - 1} = {{equ}.(32)}}}}$

Annex 8

In this annex it is proven that equ.(28) constitutes a rotation matrixand that the expressions in equ.(30) and (31) constitute a sine and acosine:

${\cos^{2W} + \sin^{2W} - 1} = {\left\lbrack {1 - \frac{\left( {u_{1} \times u_{2}} \right)^{2}}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}} \right\rbrack^{2}++}$${\left\lbrack {- \frac{\left( {u_{1}^{0} + u_{2}^{0} + {u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\sqrt{\left( {u_{1} \times u_{2}} \right)^{2}}}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)}} \right\rbrack^{2} - 1}==$$\left\lbrack {\frac{\left( {u_{1} \times u_{2}} \right)^{2}}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} - 1} \right\rbrack^{2} -$${{1^{2}++}\left\lbrack \frac{\left( {u_{1}^{0} + u_{2}^{0} + {u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\sqrt{\left( {u_{1} \times u_{2}} \right)^{2}}}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} \right\rbrack}^{2}==$$\left\lbrack {\frac{\left( {u_{1} \times u_{2}} \right)^{2}}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} -} \right.$${\left. 2 \right\rbrack\left\lbrack \frac{\left( {u_{1} \times u_{2}} \right)^{2}}{\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} \right\rbrack}++$$\frac{\left( {u_{1}^{0} + u_{2}^{0} + {u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)^{2}\left( {u_{1} \times u_{2}} \right)^{2}}{\left\lbrack {\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} \right\rbrack^{2}}==$$\underset{{{in}{general}} \neq 0}{\underset{︸}{\frac{\left( {u_{1} \times u_{2}} \right)^{2}}{\left\lbrack {\left( {{u_{1}^{0}u_{2}^{0}} + {u_{1}^{T}u_{2}} + 1} \right)\left( {u_{1}^{0} + 1} \right)\left( {u_{2}^{0} + 1} \right)} \right\rbrack^{2}}}} \cdot \cdot${[(u₁ × u₂)² − 2(u₁⁰u₂⁰ + u₁^(T)u₂ + 1)(u₁⁰ + 1)(u₂⁰ + 1)] + +(u₁⁰ + u₂⁰ + u₁⁰u₂⁰ + u₁^(T)u₂ + 1)²}

In the following we show that the second factor is zero:

[(u₁ × u₂)² − 2(u₁⁰u₂⁰ + u₁^(T)u₂ + 1)(u₁⁰ + 1)(u₂⁰ + 1)] + +(u₁⁰ + u₂⁰ + u₁⁰u₂⁰ + u₁^(T)u₂ + 1)² =  = (u₁ × u₂)²−2(u₁⁰u₂⁰ + u₁^(T)u₂ + 1)(u₁⁰ + 1)(u₂⁰ + 1) + +[u₁^(T)u₂ + (u₁⁰ + 1)(u₂⁰ + 1)]² = =u₁²u₂² − (u₁^(T)u₂)² − 2(u₁⁰u₂⁰ + u₁^(T)u₂ + 1)(u₁⁰ + 1)(u₂⁰ + 1) + +(u₁^(T)u₂)²+2u₁^(T)u₂(u₁⁰ + 1)(u₂⁰ + 1) + [(u₁⁰ + 1)(u₂⁰ + 1)]² = =u₁²u₂² − 2(u₁⁰u₂⁰ + u₁^(T)u₂ + 1)(u₁⁰ + 1)(u₂⁰ + 1) + +2u₁^(T)u₂(u₁⁰ + 1)(u₂⁰ + 1) + [(u₁⁰)² + 2u₁⁰ + 1][(u₂⁰)² + 2u₂⁰ + 1] = =u₁²u₂² − 2(u₁⁰u₂⁰ + u₁^(T)u₂ + 1)(u₁⁰ + 1)(u₂⁰ + 1) + +2u₁^(T)u₂(u₁⁰ + 1)(u₂⁰ + 1) + [u₁² + 2u₁⁰ + 2][u₂² + 2u₂⁰ + 2] = =u₁²u₂² − 2(u₁⁰u₂⁰ + u₁^(T)u₂ + 1)(u₁⁰ + 1)(u₂⁰ + 1) + +2u₁^(T)u₂(u₁⁰ + 1)(u₂⁰ + 1) + [u₁² + 2(u₁⁰ + 1)][u₂² + 2(u₂⁰ + 1)] = =u₁²u₂² − 2u₁^(T)u₂(u₁⁰ + 1)(u₂⁰ + 1) − 2(u₁⁰u₂⁰ + 1)(u₁⁰ + 1)(u₂⁰ + 1) + +2u₁^(T)u₂(u₁⁰ + 1)(u₂⁰ + 1) + u₁²u₂²+2(u₂⁰ + 1)u₁² + 2(u₁⁰ + 1)u₂² + 4(u₁⁰ + 1)(u₂⁰ + 1) = =u₁²u₂² − 2u₁^(T)u₂(u₁⁰ + 1)(u₂⁰ + 1)−2(u₁⁰ + 1)(u₂⁰ + 1) + −2(u₁⁰u₂⁰)(u₁⁰ + 1)(u₂⁰ + 1) + +2u₁^(T)u₂(u₁⁰ + 1)(u₂⁰ + 1) + u₁²u₂²+2(u₂⁰ + 1)u₁² + 2(u₁⁰ + 1)u₂² + 4(u₁⁰ + 1)(u₂⁰ + 1) = =u₁²u₂² − 2u₁^(T)u₂(u₁⁰ + 1)(u₂⁰ + 1)−2(u₁⁰ + 1)(u₂⁰ + 1) + −2(u₁⁰u₂⁰)[(u₁⁰)² + u₁⁰][(u₂⁰)² + u₂⁰] + +2u₁^(T)u₂(u₁⁰ + 1)(u₂⁰ + 1) + u₁²u₂²+2(u₂⁰ + 1)u₁² + 2(u₁⁰ + 1)u₂² + 4(u₁⁰ + 1)(u₂⁰ + 1) = =u₁²u₂² − 2u₁^(T)u₂(u₁⁰ + 1)(u₂⁰ + 1)−2(u₁⁰ + 1)(u₂⁰ + 1) + −2[u₁² + 1 + u₁⁰][u₂² + 1 + u₂⁰] + +2u₁^(T)u₂(u₁⁰ + 1)(u₂⁰ + 1) + u₁²u₂²+2(u₂⁰ + 1)u₁² + 2(u₁⁰ + 1)u₂² + 4(u₁⁰ + 1)(u₂⁰ + 1) = =u₁²u₂² − 2u₁^(T)u₂(u₁⁰ + 1)(u₂⁰ + 1) − 2(u₁⁰ + 1)(u₂⁰ + 1)+−2[u₁²u₂² + u₁²(1 + u₂⁰) + u₂²(1 + u₁⁰) + (1 + u₁⁰)(1 + u₂⁰)] + +2u₁^(T)u₂(u₁⁰ + 1)(u₂⁰ + 1) + u₁²u₂²+2(u₂⁰ + 1)u₁² + 2(u₁⁰ + 1)u₂² + 4(u₁⁰ + 1)(u₂⁰ + 1) = 0

Annex 9

In this annex the equivalence of equ.(34) and (35) is proven

${\overset{\circ}{R}(\tau)} = {{{equ}.(34)} = {{{\begin{pmatrix}{\frac{u^{0}}{\sqrt{u^{2}}}u} & {- \frac{u \times \left( {u \times a} \right)}{\sqrt{\left\lbrack {u \times \left( {u \times a} \right)} \right\rbrack^{2}}}} & \frac{u \times a}{\sqrt{\left( {u \times a} \right)^{2}}}\end{pmatrix}^{T}{\left( {1 - {\frac{1}{\left( u^{0} \right)^{2}}{uu}^{T}}} \right) \cdot \cdot \begin{pmatrix}\frac{a}{\sqrt{a^{2}}} & {- \frac{\left( {{u^{0}a} - {a^{0}u}} \right) \times \left( {u \times a} \right)}{\sqrt{{a^{2}\left( {u \times a} \right)}^{2}}}} & \frac{u \times a}{\sqrt{\left( {u \times a} \right)^{2}}}\end{pmatrix}}}=={\left\lbrack {\begin{pmatrix}{\frac{u^{0}}{\sqrt{u^{2}}}u^{T}} \\{- \frac{\left\lbrack {u \times \left( {u \times a} \right)} \right\rbrack^{T}}{\sqrt{\left\lbrack {u \times \left( {u \times a} \right)} \right\rbrack^{2}}}} \\\frac{\left( {u \times a} \right)^{T}}{\sqrt{\left( {u \times a} \right)^{2}}}\end{pmatrix} - {\frac{1}{\left( u^{0} \right)^{2}}\begin{pmatrix}{u^{0}\sqrt{u^{2}}} \\0 \\0\end{pmatrix}u^{T}}} \right\rbrack \cdot \cdot \begin{pmatrix}\frac{a}{\sqrt{a^{2}}} & {- \frac{\left( {{u^{0}a} - {a^{0}u}} \right) \times \left( {u \times a} \right)}{\sqrt{{a^{2}\left( {u \times a} \right)}^{2}}}} & \frac{u \times a}{\sqrt{\left( {u \times a} \right)^{2}}}\end{pmatrix}}=={\begin{pmatrix}{\left( {\frac{u^{0}}{\sqrt{u}} - \frac{\sqrt{u^{2}}}{u^{0}}} \right)u^{T}} \\{- \frac{\left\lbrack {u \times \left( {u \times a} \right)} \right\rbrack^{T}}{\sqrt{\left\lbrack {u \times \left( {u \times a} \right)} \right\rbrack^{2}}}} \\\frac{\left( {u \times a} \right)^{T}}{\sqrt{\left( {u \times a} \right)^{2}}}\end{pmatrix} \cdot \begin{pmatrix}\frac{a}{\sqrt{a^{2}}} & {- \frac{\left( {{u^{0}a} - {a^{0}u}} \right) \times \left( {u \times a} \right)}{\sqrt{{a^{2}\left( {u \times a} \right)}^{2}}}} & \frac{u \times a}{\sqrt{\left( {u \times a} \right)^{2}}}\end{pmatrix}}} = \ldots}}$

In the following the nontrivial components (1,1), (1,2), (2,1), (2,2) ofthe matrix R̊(τ) are explicitly calculated:

$\left( {1,1} \right) = {{\left( {\frac{u^{0}}{\sqrt{u^{2}}} - \frac{\sqrt{u^{2}}}{u^{0}}} \right)u^{T}\frac{a}{\sqrt{a^{2}}}}\overset{\overset{{u \cdot a} = {{{{- u^{0}}a^{0}} + {u^{T}a}} = 0}}{\downarrow}}{=}{{{\left( {\frac{u^{0}}{\sqrt{u^{2}}} - \frac{\sqrt{u^{2}}}{u^{0}}} \right)\frac{u^{0}a^{0}}{\sqrt{a^{2}}}}=={\left( {\frac{u^{2} + 1}{\sqrt{u^{2}}} - \sqrt{u^{2}}} \right)\frac{a^{0}}{\sqrt{a^{2}}}}} = \frac{a^{0}}{\sqrt{u^{2}}\sqrt{a^{2}}}}}$$\left( {2,1} \right) = {{{- \frac{\left\lbrack {u \times \left( {u \times a} \right)} \right\rbrack^{T}}{\sqrt{\left\lbrack {u \times \left( {u \times a} \right)} \right\rbrack^{2}}}}\frac{a}{\sqrt{a^{2}}}} = {{\frac{\left( {u \times a} \right)^{2}}{\sqrt{{u^{2}\left( {u \times a} \right)}^{2}}}\frac{1}{\sqrt{a^{2}}}} = \frac{\sqrt{\left( {u \times a} \right)^{2}}}{\sqrt{u^{2}}\sqrt{a^{2}}}}}$$\left( {1,2} \right) = {{{- \left( {\frac{u^{0}}{\sqrt{u^{2}}} - \frac{\sqrt{u^{2}}}{u^{0}}} \right)}u^{T}\frac{\left( {{u^{0}a} - {a^{0}u}} \right) \times \left( {u \times a} \right)}{\sqrt{{a^{2}\left( {u \times a} \right)}^{2}}}}==}$${{- \left( {\frac{u^{0}}{\sqrt{u^{2}}} - \frac{\sqrt{u^{2}}}{u^{0}}} \right)}\left( {u \times a} \right)^{T}\frac{u \times \left( {{u^{0}a} - {a^{0}u}} \right)}{\sqrt{{a^{2}\left( {u \times a} \right)}^{2}}}}==$${{- \left( {\frac{u^{0}}{\sqrt{u^{2}}} - \frac{\sqrt{u^{2}}}{u^{0}}} \right)}\frac{{u^{0}\left( {u \times a} \right)}^{2}}{\sqrt{{a^{2}\left( {u \times a} \right)}^{2}}}} = {- \frac{\sqrt{\left( {u \times a} \right)^{2}}}{\sqrt{u^{2}}\sqrt{a^{2}}}}$$\left( {2,2} \right) = {{{\frac{\left\lbrack {u \times \left( {u \times a} \right)} \right\rbrack^{T}}{\sqrt{\left\lbrack {u \times \left( {u \times a} \right)} \right\rbrack^{2}}}\frac{\left( {{u^{0}a} - {a^{0}u}} \right) \times \left( {u \times a} \right)}{\sqrt{{a^{2}\left( {u \times a} \right)}^{2}}}}=={\frac{\left( {{u^{0}a} - {a^{0}u}} \right)^{T}}{\sqrt{{a^{2}\left( {u \times a} \right)}^{2}}}\frac{\left( {u \times a} \right) \times \left\lbrack {u \times \left( {u \times a} \right)} \right\rbrack}{\sqrt{\left\lbrack {u \times \left( {u \times a} \right)} \right\rbrack^{2}}}}=={\frac{\left( {{u^{0}a} - {a^{0}u}} \right)^{T}}{\sqrt{{a^{2}\left( {u \times a} \right)}^{2}}}\frac{{u\left( {u \times a} \right)}^{2}}{\sqrt{\left\lbrack {u \times \left( {u \times a} \right)} \right\rbrack^{2}}}}=={\frac{\left( {{u^{0}a^{T}u} - {a^{0}u^{2}}} \right)}{\sqrt{{a^{2}\left( {u \times a} \right)}^{2}}}\frac{\left( {u \times a} \right)^{2}}{\sqrt{{u^{2}\left( {u \times a} \right)}^{2}}}}} = {\overset{{u \cdot a} = {{{{- u^{0}}a^{0}} + {u^{T}a}} = 0}}{=}{{\frac{\left( {{u^{0}u^{0}a^{0}} - {a^{0}u^{2}}} \right)}{\sqrt{a^{2}}}\frac{1}{\sqrt{u^{2}}}} = \frac{a^{0}}{\sqrt{u^{2}}\sqrt{a^{2}}}}}}$

Using these components the matrix R̊(τ) can now be written in matrix formby supplementing the trivial components (1,3), (2,3), (3,3), (3,1),(3,2):

$\ldots = {\begin{pmatrix}\frac{a^{0}}{\sqrt{u^{2}}\sqrt{a^{2}}} & {- \frac{\sqrt{\left( {u \times a} \right)^{2}}}{\sqrt{u^{2}}\sqrt{a^{2}}}} & 0 \\\frac{\sqrt{\left( {u \times a} \right)^{2}}}{\sqrt{u^{2}}\sqrt{a^{2}}} & \frac{a^{0}}{\sqrt{u^{2}}\sqrt{a^{2}}} & 0 \\0 & 0 & 1\end{pmatrix} = {{equ}.(35)}}$

The following paragraph lists all cited documents:

BIBLIOGRAPHY

-   [1] Donald E. Fahnline, A covariant four-dimensional expression for    Lorentz trans-formations, American Journal of Physics, Vol. 50,    Issue No. 9, pp. 818-821, September 1982,    https://doi.org/10.1119/1.12748-   [2] Abraham A. Ungar, THOMAS ROTATION AND THE PARAMETRIZA-TION OF    THE LORENTZ TRANSFORMATION GROUP, Foundations of Physics Letters,    Vol. 1, No. 1, pp. 57-89, 1988,    https://link.springer.com/article/10.1007/BF00661317-   [3] Howard Percy Robertson, Wigner W. Noonan, Relativity and    Cosmology, 1968, W. B. Saunders company, Philadelphia, London,    Toronto Library of Congress catalog card number 68-23690-   [4] Eric Gourgoulhon, Special Relativity in General Frames-From    Particles to As-trophysics, 2013, ISBN 978-3-642-37276-6 (eBook),    Springer Berlin Heidelberg-   [5] Oliver Davis Johns, Analytical Mechanics for Relativity and    Quantum Mechan-ics, 2nd edition published 2011, first published in    paperback 2016, Oxford University press, ISBN 978-0-19-876680-3    (PBK)-   [6] Dennis S. Bernstein, Scalar, Vector, and Matrix Mathematics:    Theory, Facts and Formulas, Revised and Expanded Edition, 2018,    Princeton University Press, ISBN 9780691151205 (hardcover), ISBN    9780691176536 (paperback)-   [7] John David Jackson, Classical Electrodynamics, Third Edition,    John Wiley & Sons, 1998, ISBN 978-0-471-30932-1-   [8] Michael Tsamparlis, Special Relativity-An Introduction with 200    Problems and Solutions, 2019, ISBN 978-3-030-27346-0, 2nd edition,    Springer Nature Switzerland AG [9] Riad Chamseddine, Vectorial Form    of the Successive Lorentz Trans-formations. Application: Thomas    Rotation, Foundations of Physics, Volume 42, Issue 4, pp. 488-511,    2012, published online Feb. 12, 2011,    https://link.springer.com/article/10.1007/s10701-011-9617-5-   [10] W. L. Kennedy, Thomas rotation: a Lorentz matrix approach,    EUROPEAN JOURNAL OF PHYSICS, Vol. 23, 2002, pp. 235-247, published    27 Mar. 2002,    https://iopscience.iop.org/article/10.1088/0143-0807/23/3/301-   [11] Leehwa Yeh, Wigner rotation and Euler angle parametrization,    22.06.2022, https://doi.org/10.48550/arXiv.2206.12406-   [12] Bernhard Strohmayer, Expression for the four-dimensional    Frenet-Serret frame of a given timelike worldline in Minkowski    space, which expression encompasses the cases that the third    4D-curvature (second torsion, hypertorsion or bi-torsion) and    possibly also the second 4D-curvature (first torsion) and possibly    also the first 4D-curvature are vanishing, U.S. patent application    Ser. No. 17/810,603 as originally filed, which can be downloaded    under https://patentcenter.uspto.gov/applications/17810603/ifw/docs    (pdf-link in the line with the document description    ‘specification’). We do not refer to the application as published by    the USPTO (publication number US 2023-0004623 A1), which is riddled    with mistakes.

1. Method of calculating for two given restricted Lorentz transformationmatrices ^(A)Lη and ^(B)Lη with identical first column vectors therotation matrix Rη defined byRη=(^(A) Lη)^(−1B) Lη, characterised in that the matrices arepartitioned.
 2. Method as defined in claim 1 in that equ.(25) is used.3. Method as defined in claim 2 characterised in that the method is usedto determine a Wigner rotation matrix (Thomas rotation matrix). 4.Method as defined in claim 2 characterised in that it is used todetermine the rotation matrix linking two different local frames of thesame timelike worldline in 4-dimensional Minkowski space.